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TEANSACTLIONS
OF THE
CAMBRIDGE
PHILOSOPHICAL SOCIETY.
ESTABLISHED Novemser 15, 1819.
VOLUME THE FOURTH.
CAMBRIDGE:
PRINTED BY J, SMITH, PRINTER TO THE UNIVERSITY;
AND SOLD BY J. & J. DEIGHTON, AND T. STEVENSON, CAMBRIDGE ; AND T. CADELL, STRAND, LONDON,
M.DCCC.XXXIII.
‘ce ; © .
t
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CONTENTS OF THE FOURTH VOLUME.
Parr I. PAGE Ne. Primriz Faune et Flore Madere et Portus Sancti; sive Species quedam Nove vel hactenus minus rite cognite Animalium et Plantarwm in his Insulis degentium breviter descripte: by R. T. Lowk, Esq.............0ee cee eee eee I
II. On the General Equation of Curves of the Second Degree: by Auaustus De Morean, Esq., Professor in the University of London................-. 71
III. On the Nature of the Light in the two Rays produced by the Double Refrac- Honvoje Quartz: (oy) Professor AURY2h\ +e sea teioie ae sie!e silesloctan(l torte ne «are 79
IV. On the Resolution of Algebraical Equations: by the Rey. R. Murpuy......... 125
V. Mathematical Exposition of some of the Leading Doctrines in Mr. Ricardo's *« Principles of Political Economy and Taxation: by the Rev. W. WHEWELL 155
Addition to a Paper “Qu the Nature of the Light in the Fwo Rays produced by the Double Refraction of Quartz:” by Professor AIRY....--....-.++++5- 199 Part II.
VI. Desoription of Chiasognathus Grantii, a new Lucanideous Insect forming the type of an undescribed Genus, together with some brief Remarks upon its Structure and Affinities. In a Letter addressed to one of the Secretaries: by Jn LP RIESE, TDG 8) ass oor docano BuO DACH BO OADCOponOnUGouOGCODDDEOUCDaOo Aut:
VII. A case of Human Monstrosity, with a Commentary: by the Rev. W. Crarx, M.D. 219
VIII. On the Examination of a Hybrid Digitalis: by the Rev. Professor HENstow...... 257
IX. Ona Remarkable Modification of Nemton’s Rings: by Professor A1ry........+..- 279
X. A Monograph on the British species of Cyclas and Pisidium: by the Rey. L. JENYNS 289
XI. On anew Analyzer, and its Use in Experiments of Polarization: by Professor Airy 313
vl
CONTENTS. Part III. PAGE On the Mechanism of the Larynx: by the Rev. R. WILLIS......--++ 0 seer ene 323 On the Inverse Method of Definite Integrals, with Physical Applications: by he REV. MR AIMED SRB EE Maeeneye tote tere ore isaice] ote ancterseta latest are nia, s'sie s[olnfors\sinja\e\s¥rerate 353
On the Phenomena of Nenton's Rings when formed between two transparent Substances of different refractive Powers: by Professor A1ry .........-...-. 409
Description of a Machine for resolving by Inspection certain important Forms of Transcendental Equations: by Sir J. F. W. Herscuer, K.G.H., &c..... 425
Pawn Ss AC TPONS
CAMBRIDGE
PHILOSOPHICAL SOCIETY.
Vou. IV. Parr I.
7 pa ay
_ ‘ee ot 7
os. : Ne ie Oe PS pee aur 1 a sf IVA LOeO * 9 a —— whe ie Lai
I. Primitie Faune et Flore Madere et Portus Sancti ; sve Species quedam Nove vel hactenus minus rite cognite Animalium et Plantarum in his Insulis de- gentium breviter descripte.
Curante Ric. Too. LOWE, A.M.
COLL: CHR. CANT. ET NUPER AB EADEM UNIVERSITATE BACC. PERIGR. [Read Nov. 15, 1830.]
ScrenTi# naturalis fautoribus haud quodammodo inutile futurum speravi, (obstantibus multis quominis Prodromum meum Faune et Flore Maderensis jam jamque edere possim), si spe- cilerum novarum vel hactents mints cognitarum selecta quadam characteribus brevibus statim exprimere curem: adjectis annotati- unculis quibusdam, supervacaneis autem omnibus excisis, prout brevitati vel maximé consulenti oportet.
Opusculum itaque de quo nune agitur, quasi Prodromus Prodromi, characteribus constat specificis rerum sine ordine et omnin6o ex arbitrio selectarum. Multa etiam nova omissa: plu- rima incerta alteri diei studioque accuratiori relicta sunt. Hee preecipue de re Entomologica et Ichthyologicé praemonenda velim : quippe ope in Insectis describendis, qué prorsus confisus eram, a morte amici C. Heineken M.D., scrutatoris vel oculatissimi, orbatus sum; itaque rem omnem (in tanta specierum novarum difficultate, entomologico vel peritissimo rem momenti non levis), radicitis suscipere insolitus inusitatusque cogor. De Molluscis, quim omnium terrestrium hactenis 4 me repertaruin catalogum completum exhibere curavi, marinas feré omnes in preesens
Vol. IV. Part I. A
2
omisi. De plantis Acotyledoneis veris (Cellularibus), de Crusta- ceis, Zoophytis, &c. idem est pradicandum. Plurima denique dubia vel nondum satis explorata in partibus omnibus consulté omisl.
Quod si in speciebus tam zoologicis quam botanicis novis rité definiendis, hac aliquantulum valeant, fructum inde percipere, haud parvum, spero; ex emendationibus, scilicet, amicis consiliis- que omnium scientiz naturalis fautorum. Rei quidem herbarize cultoribus maximé precor ut mihi quasi PoravopAw potiis quam Boravocopy indulgeant. Ommnes denique rogo, quod si in aliis corrigendis nimis aliquando videar audax, arctissimo tamen veri- tatis natureque studio me semper pro viribus eniti credant: omni petulantid mutationisque vane proclivitate ab animo longé amota.
Amicis tantis tanta debenti, singulos enumerare, sua cuique ascribenti, locus jam non adest: nec tamen omnes silentio pra- termittere possum. Cl. Rob. Brown, adjuvante J. I. Bennett arm’., summa humanitate ac benevolentia, plantarum Maderen- sium 4a Masson aliisque lectarum, in Herbario Banksiano conser- vatarum, necnon Manuscriptorum ipsius Solandri, copiam fecit. Synonymiam certissimam Helicum a cl. Sowerby, Wood, &c. descriptarum, ex autopsid speciminum ipsorum, partim in Museo Britannico repostorum, benevolentize cl'. Children, Gray, et G. B. Sowerby ipsius debeo. Amicus M. J. Berkeley characteres plan- tarum plurium novarum ab ipso in Britannid cultarum, propriis observationibus confirmavit, et iconibus accuratissimis illustravit. Cl. Hooker litteris et amicitiz que quantaque non debeo! So- cietati denique huic illustrissimz, si quid utilitatis, si quid com- modi Scientiz fautores ex his laboribus (vel minimum) haurire possint, gratias liceat omnium simul cum meis offerre: quippe que primim inceptis nostris eximid liberalitate afflavit, eorum primitias illa jam pritis inde percipere debet.
Pars [,
ie aN i a:
VASCULARES, D.C.
Cuass: II: MONOCOTYLEDONES, A. CRYPTOGAME.
Orp. I. FILICES.
Gren. ASPIDIUM, RAR. Br.
1. Aspiprum falcinellum, Sw.
A. fronde pinnata: foliis falcato-ensiformibus, acuminatis, arguté serrulatis, coriaceis, rigidis, petiolatis, basi surstm obtusé auriculatis: soris biserialibus, approximatis, distinctis: indusiorum margine stellatim dentato: stipite squamoso rhachique hirsutis.
Aspidium falcinellum, Sw. Syn. Fil. pp. 46, 243.
Hab. in cacuminibus montium. Madere; locis potissimim umbrosis, frigidis gaudens.
Cum 4. auriculato, trapexoide, Sw. &c. species a plurimis confusa. Indusia magna, orbiculata, pulcherrima. Sori conferti, aquidistantes, haud confluentes. Nomen synonymiamque cl. Swartzii hane speciem olim describentis, pro novo in MSS. imposito, retinendum curavi: mo- nentibus primtim amicis cl. Rob. Brown et J. I. Bennett armig*; quo- rum indicatione rem totam, ex autopsia speciminum in Herbario Banksiano asservatorum, luce clariorem demonstratam habeo. Swartzio A. falcinelli sui locus natalis plané latuit: quaproptér species Swart- ziana omnibus valdé dubia; nec minis quam stirps maderensis cum aliis quibusdam 4 plurimis confusa hucusqie visa est.
6 Mr. Lowe on the New Plants and Land Mollusca 2. <Aspidium frondosum, Prodr. MS.
A. fronde triangulari, tripinnata, subtts hirto-paleacea, ramosa, ramis inferioribus adscendentibus: foliolis (tertii ordinis) oblongis ova- tisque, acutis, inciso-dentatis, basi pinnatifidis, laciniis imis obtusis, den- ticulatis, ima exteriore (superiore) majore; superioribus acutis, sub-mu- cronatis: soris in foliola biserialibus, demtiim confluentibus: indusiis confertis, pellucidis, adpressissimis, planis, demim marginibus reflexis: rhachibus stipiteque basi hirsutissimo pallidis, paleaceo-hirtis.
Polypodium frondosum, Sol. MLSS’!
Hab. in sylvis rupibusque siccis umbrosis Madere: sterile haud in- frequens; cum fructu rariss. :
Frondes nitide, lucide: fertiles supra granulate s. tuberculate. Indusia in vivo squamiformia, orbiculata, maxima, membranacea, albida,
elegantia. Longit. cum stipite 2—4 pedes; Lat. 9 poll. ad 14 feré pedum. Stipes 1—2 pedalis.
3. Aspidium? drepanum, Sv.
A? fronde lanceolata, acuminata, bipinnata: pinnis acuminatis, cur-
' vyato-adscendentibus, remotiusculis: foliolis angustis, acuminatis, sub-fal-
catis, arguté inciso-serrulatis; inferioribus sub-oppositis; infimo superiore valdé elongato, rhachi parallelo; summis confluentibus: soris minutis, confertis, distinctis, biseriatis: rhachibus stipitibusque densé paleaceis.
Aspidium drepanum, Sw. Syn. Fil. pp. 54, 255.
Hab. in convallium umbrosis Madere; rariss.
Frondes rigidiuscule, 2—3 pedales; pinnis 5—6 pollices, foliolis 1 pollicem feré longis. Frondes_ steriles dissimillime, pinnis basi tan- tium pinnatis, foliolisque oblongis, multd latioribus, foliosis.
Indusia non vidi. Ob fructum in omnibus exemplaribus nimis maturum, de genere dubitandum; sc. an species Polypodiis, Nephrodivs, an Aspidiis rité consocianda. Indusia tamen, ut in <Aspidio caduco, Hook. et Grev. Ic. t. 171. minutissima, vel citd caduca esse suspicor.
of Madera and Porto Santo. 7
Gen. NEPHRODIUM, R. Br. 4. Nephrodium focenisecii, Prodr. MS.
N. fronde triangulari vel ovata, 3—4 pinnatifida, utrinque glabra: laciniis (tertii 4-tique ordinis) oblongis, obtusis; ultimis incisis, mucro- nato-serratis; omnium inferioribus exterioribus internis Oppositis majo- ribus: soris numerosis distinctis: indusiis primd semiovatis vel renifor- mibus, demtm orbiculatis, emarginatis: stipite breviusculo, basi spar- sim sub-paleaceo, fusco, superné rhachique pallidis.
a. dlatum; fronde 4—pinnatifida; pinnis inferioribus (1™ 2*° ordinis) triangularibus vel ovatis, externis interioribus oppositis valdé majo- ribus: pari infimo pinnarum (1™ ordinis) basi deorstim ramoso; pinnula (2 ordinis) potissimim 1™ (aliquando etiam 2%) inferiore s. exteriore deorstim producta.
Hab. in sylvis Vaceinii padifolii, Sm:, Madere; ubique vulgatissima.
B. productum; fronde tripinnatifidé, paulld magis elongata: pinnis om- nibus oblongis; externis internis oppositis vix majoribus: lacini- arum ultimarum dentibus sub-aristatis.
Hab. in umbrosis humidioribus Mader; rariss:
8. Status potits prioris (a), é loco obscuriore, defectu luminis, &c. quam varietas videtur.
Frons in utraque varietate nana, 1—1} pedes (una cum stipite) longa, feré pedalis; 6—8 pollices lata: stipite vix dimidium totius lon- gitudinis equante. In utraque odor idem gratissimus, foenum novum redolens, constans.
Species Aspidio dilatato et spinuloso Auct. certe proxima; et cum illis forsan, in unam speciem (ut ab amiciss. cl. Hookero) consociatis, olim conjungenda. Sed distingui posse credo, figura frondis abbrevi- ata, deltoided; stipite breviore, minus (sc. basi tanttim) paleaceo; pin- nulis angustioribus; odore. His adde frondem magis decompositam : quamvis enim rard, sc. in 8, certé mints quam in a, decomposita, in
8 Mr. Lowe on the New Plants and Land Mollusca
utroque tamen statu saltém sub-ripinnata, et longé frequentitis, sc. in a, statu normali, sub-guadripinnata*, Hee omnia, cum aliis characteri- bus supra indicatis, millibus exemplaribus stabilita sunt; et in planta a, aded per totam Insulam pervulgata, constantia, nec in tanta differentia loci coelique (8. enim potitss monstrosa) variantia inveni.
Gen. ASPLENIUM, Linn. Spr. &c.
5. Asplenium anceps, Sol. ILS.
A. fronde pinnata, lineari-lanceolata: pinnis distinctis, sub-petio- latis, oblongis, obtusis, apice sub-crenulatis, basi abruptis inferioribusque sursim acuté auriculatis: soris biseriatis, obliquis, distinctis: rhachi stipiteque nitidis, fuscis, trigonis, alato-marginatis.
Asplenium anceps, Sol. MLSS'!
Hab. in Madera, vulgaris; Asplenii Trichomanis locum tenens.
Gen. GYMNOGRAMMA, Desv. Hook.
6. Gymnogramma Lovei, Hook. et Grev.
G. fronde pinnata, utrinque hirsuta: pinnis oblongis, acuminatis, pinnatifidis ; summis confluentibus: laciniis ovalibus oblongisve, obtu- sissimis, integerrimis: stipite sparsim squamoso rhachique hirsutis.
Gymnogr: Lovei, Hook. et Grev. Ic. Fil. t. 89!
Gymnogr: Totta, “Schlechtend: (Polypod: tottum, Willd.) Spr. Syst. IV. 1. p. 38. No. 6?
Acrostichum pilosum, So/. MSS. et Herb. Banks!
Hab. in Madere umbrosis humidioribus.
Pinne infime brevitér petiolate; mediz sessiles; summz conflu-
entes: inferiores remotiores.
* Ob lacinias omnium ordinum superné confluentes, rectils 3—4 pinnatifide frondes scribuntur. Tamen ob tenuitatem suam 8—4 pinnate apparent.
of Madera and Porto Santo. 9
Orv. Il. LYCOPODIE. Gen. LYCOPODIUM, Linn., Spr.
7. Lycopodium suberectum, Prodr. MS.
L. (capsulis axillaribus): caule erecto, dichotomo; basi incurvo, decumbente: ramis fastigiatis: foliis squarrosis, 11—12 fariam imbri- catis, lineari-lanceolatis, acuminatis, rigidis, sub-pungentibus; inferioribus reflexis; superioribus erecto-patentibus.
Hab. in salebrosis fissurisque rupium sylvarum Madere.
Rami conferti, cespitosi, strictissimi, recti, crassitie digiti minim, 19 ad 16 pollices alti. Species L. Selagini proxima; sed preter alia, longé major. Inter hane et ZL. avillare Roxb. et L. Saururum Lam. et Bory, quodammodd intermedia: 4 L. Selagine tamen habitu _potis- simtm distincta. Characteribus L. rigido Sw. potitis accedit, sed ha- bitu omnino alieno. ;
B. PHANEROGAM .
Orb. ITI. GRAMINEZ. Gen. AIRA, Sm.
8. Aira argentea, Prodr. MS.
A. cespitosa: panicule coarctate, apice nutantis, ramis verticillatis, scabris: flosculis calycem equantibus, basi pilosis: aristé subdorsali, sc. ad imum feré valve nascente, recta, flosculos duplo excedente: foliis conduplicatis, filiformibus, compressis.
Hab. in Madere sylvis salebrosis.
Gren. FESTUCA, Spr. 9. Festuca Donax, Prodr. MS.
F’. panicule large, diffuse, subsecunde, nutantis ramis elongatis, flexuosis: spiculis 3-floris, lineari-lanceolatis, glomeratis; flosculis glabris, Vol. IV. Part I. B
10 Mr. Lowe on the New Plants and Land Mollusca
linearibus, muticis: foliis omnibus planis, elongatis, acuminatis; margi- nibus scabris: culmo vaginisque glabris: ligula exserta, ovata, acuta: radice fibrosa, perenni.
Hab. in Madere convallibus.
Gramen giganteum, 3—4 pedale, sylvaticum.
10. Festuca albida, Prodr. MS.
F’. densé cespitosa: panicule lanceolata, elongate, contracte, erec- tiuscule rhachi ramisque pubescentibus: spiculis puberulis bifloris; flos- culis calyce longioribus, muticis: foliis conduplicatis, elongatis, scaber- rimis, serrulatis: culmo superné vaginisque pubescentibus; ligula ab- breviata: vaginarum oris ciliatis: radice perenni.
Hab. in Maderz convallibus.
Sylvatica, bipedalis. Culmorum bases rudes, crassissimi, perennes, glomerato-cespitosi. Folia culmos sub-equantia, numerosa. Panicula pallida, albida. Spicule cum rudimento pedicellato flosculi tertii.
Habitus omnind Festuce. .
Orv. IV. CYPERACEE. Gren. CAREX, Linn., Spr.
§§. Spicis plurimis; lateralibus androgynis, pedunculatis; terminali mascula. 11. Carex myosuroides, Prodr. MS.
C. spicis ¢ (apice masculis) sub-septenis, remotissimis, solitarlis, cy- lindricis, utrinque attenuatis, densifloris, demtm pendulis, gracilibus, elongatis, simplicibus, inferioribus pedunculatis; pedunculo vaginam duplo excedente: stigmatibus tribus: fructibus, levibus, minimis, tri- quetro-oblongis, squamas lanceolatas, acuminatas «quantibus; rostro brevissimo, obtuso, integro, spits incurvo: culmo triquetro, levi.
Hab. in Madere ora septentrionali; ad margines rivulorum, sca- turigines, &c,
of Madera and Porto Santo. rf
12. Carex elata, Prodr. MS.
C. spicis ¢ (apice ¢) sub-senis, remotissimis solitariis, linearibus, laxifloris, pendulis, gracilibus, elongatis, basi compositis, ramosis, omni- bus inclusé pedunculatis sc. vagina pedunculum sub-excedente: stigma- tibus tribus: fructibus costatis, obovato-triquetris, rostratis, squamas oblongo-ovatas, aristatas sub-equantibus; rostro tenui, recto, bifido, levi: culmo triquetro, levi.
Hab. in Maderz convallibus umbrosis; sylvatica.
Orv. V. ASPARAGE. Gren. ASPARAGUS, Linzn., Spr. 13. Asparagus scoparius, Prodr. MS.
A. caule frutescente, inermi, erecto, virgato ramisque patentibus, tere- tibus, levibus: foliis fasciculatis, erecto-patentibus, teretibus, setaceis, levibus, sub-mucronatis, sub-pungentibus: pedunculis densé fasciculatis, foliis sub-brevioribus.
Hab. in rupibus Madere.
14. Asparagus scaber, Prodr. MS.
A. caule frutescente, inermi: ramis patentissimis, subdeflexis: foliis fasciculatis, rigidis, pungentibus, patentissimis, spe deflexis, ramulisque inequalitér angulatis, scabris: floribus fasciculatis; pedunculis foliorum dimidium zquantibus.
Hab. in Madere rupibus.
Gren. RUSCUS, Linn., Spr. 15. Ruscus Hypophyllum, Linn. a. latifolius ; foliis ovalibus, latioribus, 7—nerviis, distichis. R. Hypophyllum, Bot. Mag. ¢t. 2049.
Hab. in Maderz convallibus umbrosis. B2
12 Mr. Lowe on the New Plants and Land Mollusca
8. lanceolatus; foliis lanceolatis, angustioribus, numerosis, 5—nerviis; inferioribus verticillatis; caule elatiore. Hab. in convallibus umbrosis Madere. Species forsan. Caules 2—3 pedales, superne foliosi. Folia 9—16, inferiora 5—6_ pollices longa, 2—24 lata; superiora disticha. Verti- cillus imus 3—6 folius. Cetera feré ut in a.
Orv. VI. SMILACE. Gren. SMILAX, Linn., Spr. 16. Smilax pendulina, Prodr.. MS. iS. caule fruticoso, scandente, sub-aculeato, tereti: aculeis caulinis raris, sparsis, abbreviatis, deflexis: foliis inermibus, coriaceis, rigidis, un- dulatis, 7—9 nerviis, venoso-reticulatis, laté cordatis, acuminatis; peti- olis compressis, supra canaliculatis, basi 2-cirriferis: racemis flexuosis, geniculatis, longissimis, filiformibus, pendulis, terminalibus, paniculatis. ramosis; floribus ad genicula fasciculatis: baccis subglobosis, “rubris.” Smilax latifolia, So/. ISS. et Herb. Banks! non R. Br. Hab. in rupibus Madere.
Flores albi, racemis elegantissimis dispositi; feminei masculis paullo majores. Baccas rité maturas non vidi; rubras incole ferunt.
Orv. VII. DIOSCOREE, R. Br. Grn. TAMNUS, TYourn., Juss. Tamus, Linn.
17. Tamnus edulis, Prodr. MS.
Lusitanicé, “ Norsa.”
Anglice, “ Porto Muniz Yam.”
T. foliis cordato-acuminatis, 9-nerviis: stipulis sub-nullis: racemis
elongatis: floribus sub-remotis; petalis ovalibus; stigmatibus simpli- cibus.
of Madera and Porto Santo. 13
Dioscorea sativa, Bowdich, Hac. in Mad. p. 115.
Hab. in Madera.
Radix magna, extrinsécus pallidé brunnea, intts alba, flavescens : sapore miti, edulis. Flores diceci, purpurascentes, luridi. Bacce dia- metro ;poll. elliptice, “rubra.” In parochié “Porto Muniz” dicta, Caurum versus, sola colitur: incole plantam indigenam credunt; ipse nunquam nisi plané cultam aut ex cultu ortam vidi. In Canariis, monente amico P. B. Webb, arm., procul dubio indigena; unde forsan in Maderam introducta est.
Radix in aqua multas horas (X ad XII.) bulliente, tandem coctilis.
Orv. VIII. ORCHIDE. Gen. ORCHIS, R. Br.
18. Orchis foliosa, Sol. MSS.
O. tuberibus palmatis: labello trilobo, subplano, expanso, latiore quam longo; lobo medio lateralibus rotundatis, crenulatis angustiore, obtuso, integro: sepalis obtusiusculis; exterioribus erectis; duobus in- terioribus reflexis: germine cornu descendens, tenue, zquale, obtusum superante: bracteis foliaceis, flores zquantibus: caule solido, elato.
Orchis foliosa, Sol. MSS, Masson, et Herb. Banks!
Hab. in umbrosis convallium sylvisque Madere.
Ab Orchide longibracteatd Bivon, (Bot. Reg. t. 357): quacum a nonnullis confusa, omnind distincta. Flores magni, purpurei, inodori. Caulis 2—pedalis.
Gren. GOODYERA, R. Br. 19. Goodyera macrophylla, Prodr. MS. Tab. I. ff. 1—12.
G. perianthii campanulati labello glabro cochleari-calceolato; se- palis tribus exterioribus pubescentibus: columna anticé acuminata:
massis pollinis lineari-clavatis: spica pubescente: floribus secundis, brac- teas superantibus: foliis ovalibus, reticulato-nervosis: caule repente.
14 Mr. Lowe on the New Plants and Land Mollusca
Hab. gregaria in declivibus sylvarum Madere humidis, umbrosis. Rariss.
Caules repentes; demim erecti, pedales; superné cum bracteis, germinibus, sepalisque tribus exterioribus pallidé ferrugineo-pubescentes. Folia sat magna se. semipedalia, 3 poll. lata. Spica secundiflora, primim pyramidata. Flores conferti, inodori, albidi, sub-cernui, ~ poll. longi.
Crass. III. DICOTYLEDONE.
Orv. I. AMENTACEE. Gen. SALIX, Linn. Spr. * Amenta precocia.
20. Salix canariensis, Sm.
§. arborescens, ramis glaucis, pruinosis, petiolisque tomentosis : foliis lanceolatis, elongatis, utrinque attenuatis, sub-integerrimis; supra glabriusculis, lucidis; subtts glauco-incanis, sub-tomentosis: stipulis mi- nutis, adpressis, ovatis, crenatis: squamis ovato-oblongis, obtusiusculis, sub-spathulatis, sericeo-villosis: germinibus magnis, pedicellatis, ovato- lanceolatis, acuminatis, styloque abbreviato glabris: stigmate utroque demum bifido.
Hab. in rupibus madidis Madere: etiam Nivarie, P. B. Webb, arm.
Arbor feré 20 pedes alta evadit. Ramuli crassi, seepe colorati. Gemme magne. Amenta ¢ cylindrica, abbreviata, 2-andra; ¢ elon- gata, graciliora. Ex characteribus videtur S$, pomeranice Willd. affinis.
of Madera and Porto Santo. 15
GEN. QUERCUS, Linn.
21. Quercus mitis, Herb. Banks.
Q. foliis ovatis, subcordatis, obtusis, integriusculis, sinuolato-denti- culatis, dentibus remotis, obsoletis; subtus, petiolis, ramulisque incano- tomentosis.
Quercus mitis, Herb. Banks!
Hab. in “ Madera, Donne 1776:” Herb. Banks.
Ramuli (in specimine Banksiano) sub-umbellati; juniores tomento brevissimo, cinereo obducti. Petioli sub-semipollicares. Folia 1j—2 poll. longa; 1;—1} lata; alterna, ovata, obtusissima, basi sub-cordata, integriuscula s: marginibus sub-sinuolatis, nervisque lateralibus in denti- culos remotos, obsoletos excurrentibus; coriacea, venosa, venis subtiis prominentibus; supra lucida, glaberrima; subtis cum petiolis tomento brevissimo densé velutina, in junioribus albo-incano, demim sub-ferru- gineo. Flores masculi sessiles, glomerati, in spicis abbreviatis se: vix uncialibus, axillaribus, inferioribus congesti: Feminei pauciores, spicati, vel solitarii pedicellati, superiores sc: in axillis foliorum terminalium versus apices ramulorum supra masculos nascentes. Rhachis spicarum calycesque tomentosi vel lanuginosi.
Fructus in specimine deest.
Orv. II. URTICE. Gen. URTICA, Linn. Spr.
22. Urtica elevata, Prodr. MS.
U. caule suffruticoso, lignoso, foliisque oppositis, longé petiolatis, cordato-ovatis, grossé dentatis, lucidis, glabris: petiolis filiformibus sparsim setosis: spicis axillaribus, pedunculatis, filiformibus, simplicibus,
interruptis, paucifloris, laxis, folio multtm brevioribus, Urtica elevata, Herb. Banks !
Hab, in rupibus convallium Maderz.
16 Mr. Lowe on the New Plants and Land Mollusca
Rami tenues, diffusi, debiles. Folia sat magna, ad apices ramo- rum sub-conferta. Habitus omnino generis. Planta inermis (haud urens se. pungens).
Gen. PARIETARIA, Linn., Spr. 23. Parietaria gracilis, Prodr. MS.
P. lucida, pubescens: caulibus ramisque gracilibus, erectis: foliis rhombeo-ovatis, obtusis, 3-nerviis, petiolatis; petiolis filiformibus, folia equantibus: glomerulis axillaribus; floribus pedunculatis, sub-cymosis, 1—2—3 bracteatis; bracteis (seepitis 3) angustis, lanceolatis, calyce 4—fido brevioribus, post anthesin glanduloso-pubescentibus, inzequalibus, 1—2 dilatatis, foliaceis, calycem superantibus.
Hab. in Madera; rariss.
Orn. III. LAURINEE. Gren. LAURUS, Spr.
24. Laurus Barbusana, Prodr. MS. .
L. foliis perennantibus, lanceolato-oblongis, utrinque attenuatis, coriaceis, rigidis; supra nitidissimis; infra axillis venarum nudis (e-glandulosis): pedunculis ad ramulorum apices congestis, paniculatis, sub-racemosis: pedicellis sub-elongatis, laxis: floribus hermaphroditis ; calycibus sexfidis.
Hab. in Madere Sylvis.
Arbor magna. Folia sepe cymbiformia. Anthere biloculares. Drupa ? pollicis longa, 4 lata; non calyculata.
Orv. IV.§ CHENOPODIE. Gen. ATRIPLEX, Linn., Spr. 25. Atriplex parvifolia, Prodr. MS.
A. suffruticosa, procumbens, farinoso-incana: foliis confertis, alter- nis, ellipticis vel oblongis, repandis, sub-sinuato-erosis vel integris: valvis hastatis, integerrimis, dorso muriculato-tuberculatis.
of Madera and Porto Santo. 17
Atriplex portulacoides (angustifolia) Herb. Banks!
Atriplex portulacoides var. angustifolia, So/. MSS’!
Hab. in Insula Portis S". In Canariis, P. B. Webb, armig.
Species videtur: Cum A. portulacoide vera spe forsan confusa. Cf. A. portulacoiden Desf: Fl. Ail. U1. p. 392. An hue quoque spectat A. verrucifera B. angustifolia Bieberst?
Orv. V. NYCTAGINES®. Gen. MIRABILIS. Linn, Spr.
26. Mirabilis divaricata, Prodr. MSS. M. floribus congestis, terminalibus, sub-pedunculatis: corollé calycem sextuplo superante; tubo longissimo, pubescente; limbo plicato (laciniis emarginatis) tubi quartam partem aquante: foliis sub-cordatis, petiolatis ; supra, petiolis, lineaque caulina utrinque exarata sub-pubescentibus : ramis dichotomis, nodosis, cauleque erectis: pericarpio rugoso, glabro, (atro).
Mirabilis hybrida, Lepel/? sed folia in planta Maderensi (quamvis lucida) minimé glabra, &e.
Hab. in hortis et ruderatis Madera. Circa urbem Funchalensem nune quasi indigena.
Valde ramosa, 3—5 pedalis; ramis divaricatis, demum corymbosis vel convexo-fastigiatis. In M. Jalapé vera caules multo humiliores, minus ramosi; pericarpia minora, ferrugineo-pubescentia, minus rugosa,
egranulata.
Orv. VI. PLANTAGINES. Gren. PLANTAGO, Linzn., Spr. 27. Plantago leiopetala, Prodr. MS.
P. caulescens: caule abbreviato, basi frutescente: foliis confertis, lanceolatis, utrinque attenuatis, nervosis, glabriusculis, nitidis, integerri- mis: pedunculis folia superantibus, angulatis, glabris: spicis abbreviatis,
Vol. IV. Part I. (o
18 Mr. Lowe on the New Plants and Land Mollusca
oblongis ovatisve, obtusis, nudis: laciniis calycinis latis, scariosis, cari- natis, corollisque glabris. Hab. in cacuminibus Ins®: Portis 8".
Plantagini lanceolate proxima; cultura non mutatur.
Orv. VII. PLUMBAGINE. Gen. STATICE, Spr. 28. Statice pyramidata, Prodr, MS.
S. cespitosa, glauca: scapo erecto, ramoso, aphyllo: foliis radica- libus, parvis, obovato-oblongis, acutis, mucronulatis, in petiolum atte- nuatis, enerviis: panicule pyramidate ramis patentissimis, recurvis: floribus conglomerato-imbricatis; laciniis ealycinis obtusiusculis.
Hab. in rupibus maritimis Ins*: Portis 5°.
Flores pallidé cerulei, parvi, glomerulis congesti. S. awriculefolie et oleefolie affinis. A S. spathulatéd Desf. differt foliis acutis; scapo magis ramoso; ramulis gracilioribus, sub-deflexis; floribus glomerulatis, minoribus, &e.
Orv. VIII. LABIATA. Gren. SALVIA, Linn., Spr. 29. Salvia collina, Prodr. MS.
S. caule herbaceo, viscoso-piloso: calyce 5-dentato, S foliis pin- natifidis, incisis, vel sub-sinuatis, dentatis, venosis, glabris, leviusculis : bracteis sub-rotundis, latis, cordatis, abbreviatis, acutis, inconspicuis, ca- lycis dimidium xquantibus, integerrimis: verticillis 6-floris: corollis calycem dupld superantibus: galeé falcata, compressa : lobo medio labii inferioris cucullato; lobis lateralibus reflexis, parvis, abbreviatis, rotun- datis, obtusis.
Salvia verbenacoides, Brot? (polymorpha, Hoffm.)
Hab. in collibus pascuisque altis Mader.
of Madera and Porto Santo. 19
Salvie pratensi et Verbenace (potissimum priori) affinitas summa ; sat verd distincta. A Salvia bicolori Desf: differt caule humiliore; fo- liis glabris; bracteis abbreviatis, latis. acutis; lobis lateralibus _labii inf*. rotundatis, &c.
Gren. THYMUS, Linn. 30. Thymus micans, Sol. MSS.
T. pedunculis ad apices ramulorum congestis, sub-racemosis, axil- laribus, oppositis, solitariis, unifloris: calycis $ labio superiore lato, ob- soleté tridentato, marginibus recurvis; inferiore dentibus duobus sub- erectis, lanceolatis, acutis, contiguis, equalibus profundeé inciso: bracteis linearibus foliisque lineari-spathulatis, obtusis, basi attenuatis, pilisque raris, longis, patentissimis, remotis, pectinato-ciliatis: caulibus hispidis, prostratis, cespitosis, basi fruticulosis.
Thymus micans, Herb. Banks. et Sol. MSS'!
Hab. copiosissimé, cespitem efficiens, per totum campum illum excelsum (5000 ad 6000 pedes altum) “Paul da Serra” ‘dictum, Madere.
Grn. SATUREJA, Linn. 31. Satureja thymoides, So/. MSS.
S. pedunculis axillaribus, multifloris, umbellatis: bracteis setaceis, fasciculatis: dentibus calycinis dimidium tub sub-zquantibus: foliis oblongo-lanceolatis, acutis, utrinque attenuatis, margine revolutis, sub- puberulis, subttis sub-incanis: ramulis junioribus sub-pubescentibus; caule fruticuloso, erecto.
Satureja thymoides, Sol. MSS. et Herb. Banks!
An Thymus therebinthaceus, Willd. Enum. Pl. Hort. Berol. p. 6242
Hab. in Madera et Portu S$"; vulgaris.
Fruticulus elegans. Folia sub-micantia, odorata. [lores sub-con- spicui, purpurascenti-rosei. Stamina tubo corolla breviora, inclusa,
c2
20 Mr. Lowe on the New Plants and Land Mollusca
Calyces striati, sub-pubescentes; fauce villis clausa; dentibus setaceis. sub-zqualibus, duobus inf*. aliquando sub-longioribus. Folia latitudine. &e. variabilia; ideoque 7’ therebinthaceus Willd. (si ex descriptione ju- dicare licet), vix nisi genere discrepans, idem videtur.
Orv. IX. PERSONATAE. Gen. EUPHRASIA, Linn., Spr.
32. Euphrasia Holliana, Prodr. MS.
E. \aciniis calycinis, foliisque lanceolato-oblongis, obtusis; inferio- ribus grossé dentatis; summis sub-integris: corolla (lutea) calycem duplo excedente; staminibus corollam sub-zquantibus: caule ramoso.
Bartsia viscosa, var. foliis linearibus obtusis &c. Herb. Banks !
Hab. in sylvis Madere.
Corolla conspicuz, sat magne; labium inferius $-lobum, lobis ro- tundatis, obtusissimis, denticulatis; superius simplex. Folia et habitus quodammodd Euphrasie Odontitis.
Ob antheras distinctissimé aristatas, vera Huphrasie species: quin- etiam toto habitu, caule ramoso &c. 4 Bartsid viscosd differt. Ab Euphrasid luted, cui affinitate propior, dignoscitur caule ramisque ro- bustis, nec filiformibus; foliis multd latioribus majoribusque, inferiori- bus grossé vel inciso-dentatis: floribus multd majoribus, &c.
Nomen dedi in honorem amici F'r. Holl, botanosophi Germanici, indefessi plantarum Maderensium indagatoris.
Gren. SCROFULARIA, Linzn., Spr. 33. Scrofularia racemosa, Prodr. MS.
S. foliis sub-cordato-oblongis, acutis, sub-duplicato-serratis, utrinque cauleque acutangulo glabris, basi inequalibus; inferioribus appendicu-
of Madera and Porto Santo. 21
latis: thyrsi elongati aphylli ramis racemosis; racemis elongatis, flex- uosis, patentibus, laxis, ramis pedicellisque sparsim sub-glandulosis: ca- lycis glabriusculi laciniis obtusis: corolle labii inferioris lobo intermedio revoluto, vix prominulo, minuto; superioris, rudimento staminis 5°.
squameformi, plano, rotundato. Serofularia auriculata. Linn., Spr. &c. 2
a. longifolia; glaberrima, foliis acuminatis, elongatis, simplicitér serrato-
crenatis.
B. puberula; foliis radicalibus et junioribus subtis petiolisque pube- rulis.
Hab. a. et. #. ad rivulos et in rupibus madidis convallium Ma- dere.
Scrof. sulphurea Mill. Dict.; et S. Balbisii Wild. (ex Spr.), saltem ex descriptionibus, equé plant nostre a. pertinere possent. S. awricu- lata vera (cui prior forsan synonyma), foliis “ obtusis subtus hirsutis” Spr. (“tomentosis” Linn.) “lobo terminali cordato aut ovato” Desf: differre videtur. S. auriculatam Brot. Fl. Lusit. I. p. 201 vero, ob fo- lia “subtus glabra,” ab a. nostra alienam egré putarem, nisi quod a nonnullis ad S. trifoliatam Linn. relatam video. In re tam dubia, plan- tam Maderensem pro tempore distinctam servandam putavi.
34. Scrofularia hirta, Prodr. MS.
S. foliis cordato-oblongis, acutiusculis, basi sub-aqualibus, excisis, arguté duplicato-inciso-serratis, utrinque petiolis cauleque acutangulo villosis; petiolis latis, sub-alatis, ex-appendiculatis, hirtis; thyrsi aphylli ramis trichotomo-racemosis; racemis elongatis, pedunculisque glanduloso- pubescentibus; calycis glaberrimi laciniis obtusissimis; corolla labii in- ferioris lobo intermedio revoluto, vix prominulo, minuto; superioris, rudimento staminis 5". papilleformi, minutissimo, brevi; genitalibus exsertis.
Hab. in Maderz umbrosis humidis obscuris. Rariss.
22 Mr. Lowe on the New Plants and Land Mollusca
Orv. X. CONVOLVULE. Gen. CONVOLVULUS?
35. Convolvulus? solanifolius, Prodr. MS.
C? caule volubili, fruticoso: foliis cordatis, ovato-oblongis, acutis, integerrimis, petiolatis; junioribus, ramulis, petiolisque pubescentibus : pedunculis axillaribus, solitariis, petiolo longioribus, apice sub-trifloris, pedicellisque elongatis, nudis: calycibus ovalibus, obtusiusculis; .... .
Hab. in Madere rupibus. Rariss.
Corollam nondum vidi.
Orv. XI. SAPOTE. Gey. SIDEROXYLON, Spr.
36. Sideroxylon Mermulana, Prodr. MS. “ Mermulana,’ incolarum.
S. inerme: foliis obovatis, obtusis, spathulatis, integerrimis, cori- aceis, nervosis, lucidis, utrinque glaberrimis : pedunculis unifloris, ad axillas aggregatis, brevibus, calycibusque velutinis.
Sideroxylon Mermulano, Herb. Banks. !
Hab. in rupibus, presertim maritimis, Madere.
Frutex, vel sub-arboreum. Flores parvi, pallidé carnei. Fructus ruber, edulis.
Nomen “ Mermulana” in Canariis plante distinctissime, sed quoad habitum simillime, se. Myrsinet canariensi, impositum scribit amicus P. B. Webb, arm.
Orv. XII. COMPOSIT &. * CICHORACE. Gen. SONCHUS, Linn., Spr. 37. Sonchus ustulatus, Prodr. MS.
S. glaberrimus: caulibus simplicibus, brevissimis, herbaceis, basi sub-lignosis: foliis radiatim confertis, decursivé runcinato-pinnatis, sub-
of Madera and Porto Santo. 23
carnosis, rigidis, subtus prasertim inter venas pulchré glaucescentibus ; foliolis acutis, angulatis, sub-integerrimis vel dentibus sparsis, raris, mi- nutis, callosis: scapi terminalis, aphylli, pauciflori, ramis raris, divari- catis, solidis, pedunculisque 1-floris, supra incrassatis, paniculatis, nudis ; squamis anthodii purpureo-nigricantibus, adpressissimis, latis.
S. hyoserifolius, (Hornem:) Spr?
a. angustifolia; foliolis angustis, confertis, acuminatis, margine poste- riore sub-integerrimo.
Sonchus dentatus, Herb. Banks?
B. latifolia: foliolis majoribus, latioribus, distantibus, angulatis, utrinque denticulatis: foliis supra vix glaucescentibus, profundits incisis.
“ Sonchus squarrosus ” (lined atramenti per medium verborum squar- rosus (3 ducta, et “fruticosus” supra scripto) “et MSS. differt pani- cula dichotomaé—planta minor. Madera Fr. Masson.” Herb. Banks!
Hab. in rupibus maritimis aridis Madere.
B (vix var.) est status potitis € solo vel humidiore vel magis um- broso ortus.
De Soncho hyoseridifolio hue rite referendo, suspensus hereo. Characteres feré iidem; nisi qudd illa inter “ fruticosos” wmumeratur, cum nostra planta certissimé “ herbaceis” releganda est. An descriptio cl. Sprengelii & specimine manco, (se. sine radice desiccato), in her- bario servato, (quale est forsan Sonchus dentatus Herb. Banks.), facti- tata; ideoque erronea? Nam in tali, caulis casu forsan quodam ligno- sus evadere posset: specimine ttim speciem ramuli fruticis cujusdam ramos omnino prebente.
S. dentatus Herb. Banks. in omnibus nisi caule lignoso cum vari- etate a. nostra convenire videtur. Sed, ciim preter hoc, alia exstant specimina in Herb. Banks., ad (3. nostram certissimé pertinentia, que a cl. Solandro ad alteram (quamvis revera distinctissimam et longé alienam) speciem (S. squarrosum) referuntur, ideoque 4 S. suo dentato plane distinguuntur, impensits suspicandum est hune S. dentatum a
24 Mr. Lowe on the New Plants and Land Mollusca
nostra planta alienum esse, et forsan S. hyoseridifolii (Hornem.) Spr. veri synonymam. In re tam incerta, difficultatem minus nomine plané
novo quam veteri incerto augeri putavi.
Gren. TOLPIS, Gaert. 38. Tolpis crinita, Prodr. MS. Tab. Il. ff. 1—3.
T. caule ramoso: ramis virgatis: foliis radicalibus humifusis, solo
adpressis, plerumque sinuato-pinnatifidis, sub-canescentibus ; caulinis an- gustis, lanceolato-linearibus: bracteis setaceis, abbreviatis, ad apices pe- dunculorum inferné nudorum congestis: seminibus (omnibus, radii sc. conformibus) sub-quadrisetis.
Crepis crinita, Sol. MSS. et Herb. Banks!
Crepis incrassata, Herb. Banks. “Insula Azores F ayal Mess*™, Forster” !
Hab. in Madere collibus apricis. —
Gren. CREPIS, Spr. 39. Crepis pectinata, Prodr. MS.
C. caule frutescente, ramoso, foliato; ramis diffusis, virgatis: fo- liis flaccidis, tenuissimé et profundé divisis, pectinato-pinnatifidis; laci- niis distantibus, elongatis, lineari-filiformibus, supra glabris, subtus sub- farinosis: pedunculis proliferis, superné incrassatis, squamosis, squamisque minutis, erectis, anthodioque farinoso-albescentibus.
Crepis tenuifolia, Sol. MSS. et. Herb. Banks! non Willd.
Hieracium fruticosum foliis tenuissimé coronopi modo divisis. Sloan. Cat. 123.— Hist. Jam. p. 19. t. 5. f.1, 2. (Icon. mala; Deser. opt.)
Hab. in rupibus apricis Mader ubique.
Affinitate et habitu Crepidi succulente Hort. Kew: (C. coronopi- folie Desf:) proxima; cujus speciei, pre ceteris polymorphe, (in Ma- dera vix mints vulgaris), varietatem esse meram alia forsan dies do- cebit. In illé tamen, quamvis folia aliquando profundits pinnatifida
of Madera and Porto Santo. 25
quam in statu normali, nunquam ut in C. pectinatd nostra feré fili- formia et tenuitér divisa, laciniis elongatis linearibus (feré ut in Core- opside tinctorid Hort.) videntur.
Oss. C. succulenta Ait. et C. pectinata nob: genus Tolpidem cum Crepide arctissimé conjungunt.
40. Crepis macrorrhiza, Prodr. MS.
C. glaberrima: radice perenni, crassa, carnosa: caulibus solidis, fo- liatis, simplicibus, superné paniculatis: foliis omnibus indivisis, oblon- gis, dentatis, sessilibus, nitidis, sub-carnoso-coriaceis: panicula larg4, mul- tiflora; pedunculis superné sub-incrassatis, squamosis; anthodiis sub- farinoso-pubescentibus.
Crepis macrorrhiza, Herb. Banks: et Sol. MSS! Hook. in Bot. Mag. t. 2988!
Hab. in Maderz rupibus.
41, Crepis? andryaloides, Prodr. MS.
C? glanduloso-hispida: radice carnosa, bienni: caule sub-fistuloso, foliato, simplici, superné laxé paniculato, hispido: foliis omnibus in- divisis, oblongis, acuminatis, undulatis, remote runcinato-dentatis, sub- sinuatis, sessilibus, hispidis: floribus laxé paniculatis, remotis: pedun- culis nudis, gracilibus, divaricatis, laxis, anthodiisque cylindricis, glan- duloso-hirsutissimis: involucro erecto, persistente.
Hab. in convallibus Madere.
Semina matura non vidi; pappus in immaturo revera sessilis: sed cum in veris quibusdam Borkhausie speciebus pappus in semine im- maturo omnino sessilem vel subsessilem videre licet, in nostra forsan planta pappus seminis maturi stipitatus evadit.
Gen. BORKHAUSIA, Bohm. Spr. 42. Borkhausia laciniata, Prodr. MS.
B. radice annua: caule erecto, stricto, ramoso, paniculato, sub- puberulo, nitido: foliis laciniato-pinnatifidis, vel runcinato-dentatis, sinu- Vol. lV. Part I. D
26 Mr. Lowe on the New Plants and Land Mollusca
atis, glabris; radicalibus plerumque integriusculis, oblongis; caulinis lineari-lanceolatis, semi-amplexicaulibus, basi auriculatis, sub-sagittatis, dentato-laciniatis: floribus corymboso-paniculatis: anthodii squamis dorso inferné nigrescenti-glanduloso-hispidis, interstitiis sub-farinoso-puberulis : squamis involucri laxis, farinoso-puberulis.
a. pinnatifida; foliis profundius divisis. Crepis biennis, Herb. Banks! quoad specimina in Madera a Masson lecta. “Crepis Dioscoridis” (linea per verbum Dvoscoridis ducta) “ L. var. corolla undique lutea. Madeira Fr. Masson 1777.” Herb. Banks! B. integrifolia; foliis integriusculis. “Crepis Dioscoridis” (lined per verbum Dioscoridis ducta) “ L. var. foliis margine nudis, Madeira Fr. Masson 1777.” Herb. Banks!
Hab. in Madera; in vinetis, locis cultis, frequens.
43. Borkhausia divaricata, Prodr. MS.
B. yradice crassa, fusiformi, bienni (perenni?): caulibus ramosis, paniculatis, solidis, inferné glabris, superné pedunculisque divaricatis, patentibus, hispido-glandulosis: foliis rigidis, glaberrimis, undulatis; radicalibus sinuato-runcinatis, caulinis basi semi-amplexicaulibus, dila- tatis, ovato-acuminatis, integriusculis: floribus sparsis, paniculatis: an- thodiis post anthesin ovatis, basi ventricosis; squamis basi hispido- glandulosis, superné squamisque laxiusculis involucri glabris.
a. robusta; caulibus erectis, virgatis, pedalibus, miultifloris, foliosis: foliis seape runcinato-pinnatifidis.
Hab. in Promontorio S“. Laurentii Madere.
6. pumila; caulibus sepe diffusis, glabriusculis, paucifloris, plerumque nudis: foliis radicalibus indivisis, integriusculis vel runcinato-sinu- ato-dentatis, sub-carnosis: anthodiis hispidioribus.
Hab. in Portii S"°.—Status potits, ex solo aridiore, quam varietas praecedentis.
of Madera and Porto Santo. 27
44. Borkhausia hieracioides, Prodr. MS.
B. yadice annua?: caule erecto, ramoso, paniculato, foliarum costa centrali, pedunculis, anthodiisque setoso-hispidis vel sub-muricato-spinel- losis: foliis glabris, indivisis, denticulatis, denticulis raris, sparsis, mi- nutis, subulatis; radicalibus lanceolato-oblongis, acutis, basi attenuatis ; caulinis ovato-acuminatis, basi dilatato-auriculatis, semi-amplexicaulibus : floribus corymbosis; flosculorum ligulis elongatis, laxis, patentissimis, sub-pendulis.
a. integrifolia; foliis integriusculis, sub-sinuolatis.
B. pinnatifida; foliis sub-pinnato-runcinatis. Hab. in Madere ora Septentrionali.
45. Borkhausia dubia, Prodr. MS.
B. radice bienni: caule erecto, stricto, @ basi ramoso, ramisque foliatis, costaque centrali foliarum subtts hispidis: foliis lucidis, gla- bris, indivisis, marginibus undulatis, sinuato-runcinatis et denticulatis, denticulis intermediis plurimis, inequalibus, subulatis vel ciliato-seta- ceis; radicalibus elongatis, oblongo-lanceolatis, acutis, basi attenuatis; caulinis basi cordato-zqualibus, amplexicaulibus, oblongis, acuminatis ; summis linearibus, sub-integerrimis, setaceo-ciliatis: floribus sub-corym- bosis: ramulis superné, pedunculis, anthodiisque dense glanduloso-pu- bescentibus, sub-incanis, farinaceo-puberulis: pappo sub-stipitato.
Hab. in convallibus Madere.
Precedenti proxima; differt autem habitu distinctissimo, ramis superné, pedunculis, anthodiisque densé glanduloso-pubescentibus, sub- incanis, farinaceo-tomentosis; floribus minoribus in corymbos laterales collectis, floseulorum ligula nee elongata nec pendula, pappo sub-stipi- tato (quo ad Crepides veras accedit), foliarum margine inequaliter sed conspicué et in omni parte runcinato-sinuatis, caulinis basi aqualibus, cordatis, (non dilatato-auriculatis).
46. Borkhausia comata, Prodr. MS.
B. radice fusiformi, carnoso: caule erecto, é basi ramoso, foliato,
hirto-setoso: foliis indivisis, denticulatis; radicalibus glabris; caulinis D2
28 Mr. Lowe on the New Plants and Land Mollusca
summis ciliato-crinitis: floribus corymbosis: anthodiis hirsutissimis, co- matis; squamis crinitis.
Crepis comata, Herb. Banks. et Sol. MSS!
“ Hab. in Madere sylvis; Fr. Masson 1777.” Sol.
Pappus distinctissimé stipitatus.
Gen. THRINCIA, Roth, Spr.
47. Thrincia nudicaulis, Prodr. MS.
T. foliis hispidis, sub-dentato-sinuatis: pappo disci stipitato.
Leontodon nudicaule, Herb. Banks!
Hab. in apricis Madere ubique; vulgatissima.
Flosculorum tubulus ad apicem pilosus; ligulorum lacinize eglandu- lose. Pappus radii paleaceus; disci plumosus. Semina sursim atte- nuata, acuminata, in rostrum gracile, elongatum producta; unde pappus stipitatus.
Thrincia hirta Hook. Brit. Fl. (Apargia hirta Sm. Eng. Fi., He- dypnois hirta Ejusd. in Engl. Bot.) pappo disci sessili, potissimim differt. Eodem charactere, necnon genere, sc. pappo radii paleaceo, ab Apargid hispida omnind distincta.
* * CINAROCEPHALA Gren. CIRSIUM, Tourn, Spr., &e. (CNICUS, Aliorwm). 48. Cirsium latifolium, Prodr. MS.
C. inerme: foliis sessilibus, basi auriculatis, amplexicaulibus, om- nibus elliptico-oblongis, latis, obtusis, indivisis, laté sinuato-crenatis, setoso-spinelloso-ciliatis, supra lucidis, nudis, subtus cauleque lanato- tomentosis, floccosis: pedunculis longissimis, floccosis, unifloris: antho- diis sub-lanatis: squamis lanato-ciliatis, mucronatis, adpressis, inferiori- bus ovatis, acutis; superioribus oblongis, obtusiusculis.
Carduus latifolius, Herb. Banks !
of Madera and Porto Santo. 29
Hab. in Madere convallibus.
Species pulchra, distinctissima, C. heterophyllo affinis. Caulis 2—3-—pedalis. Folia ampla, subtis sepe nivea. Flores purpurel. ** * CORYMBIFER®. Gren. GNAPHALIUM, Linz., Spr. &e
49. Gnaphalium melanophthalmum, Prodr. MS.
G. fruticosum: foliis sparsis, sessilibus, lanceolatis, acuminatis, basi attenuatis, ramisque niveo-tomentosis, canescentibus: paniculis termi- nalibus, congestis, corymbosis: squamis anthodii nivei, globosi, laxis, ovatis; inferioribus obtusis, rotundatis; superioribus acutiusculis.
Gnaphalium rupestre, Herb. Banks ! Oss. Gnaph. rupestre, Rafin: jam adest. Steud. Nom. Bot. Hab. in rupibus convallium Madere.
Flores nivei, odori; disco post anthesin nigro.
Orv. XIII. RUBIACE®. Gen. GALIUM, Linn, Spr.
50. Galium productum, Prodr. MS.
G. glabrum: foliis octonis, lanceolato-linearibus, acutis, cuspidatis, reflexis, sub-integerrimis, denticulis marginalibus raris, obsoletis, antror- sim spectantibus, utrinque levibus, supra cauleque pariim ramoso lucidis: panicularum lateralium terminaliumque ramis divaricatis, ab- breviatis: corolla laciniis obtusiusculis, mucronatis: fructibus levibus, glabris: caule 4—angulari, debili, diffuso, elongato, simpliciusculo, levi, basi suffruticoso.
Hab. in Madere saxosis, sepibus, rupibus &c. frequens.
30 Mr. Lowe on the New Plants and Land Mollusca
Orv. XIV. UMBELLIFERZ. Gen. @NANTHE, Spr. 51. Qnanthe pteridifolia, Prodr. MS.
iS. radicibus tuberosis, fusiformibus, fasciculatis: caule erecto, in- ferné tereti, levi, ramis angulatis, striatis: foliis omnibus tripinnatis ; pinnis pinnulisque omnibus remotis, oppositis, patentissimis, distichis ; foliolis ultimis ovatis lanceolatisque, acutis, inciso-dentatis pinnatifidis- que, basi cuneatis: umbellis oppositifoliis; radiis inaequalibus; bracteis paucis, subnullis, bracteolisque linearibus: fructu suberoso.
CE. apiifolia, Brot?
Hab. in rupibus madidissimis convallium Madere.
Radices repentes; tuberibus fusiformibus, fasciculato-filipendulis, erassitie digiti. Caules elati, fistulosi, esculenti. Folia maxima, ele- gantia, latevirentia, foliolis exiguis, tenuibus, concinnis. Unmbelle me- diocres, sat parve; floribus albidis, aspectu eorum -dinanthes crocate. Calyx persistens. Petala mucrone elongato, inflexo. “ Floral Recep- tacle,” Sm. nullum. Stylopodia (* Bases of Styles,” Sm.) tumida, glo- bosa. S*yli persistentes, post anthesin elongati, fructum maturum equantes. Fructus ovato-oblongus, lateralitér (sc. sutura) compressus, presuberosus. Mericarpia*, striis 7 dorsalibus, levibus, sub-zqualibus, tribus vix majoribus; interstitiis angustis, planis, equis; sutura utrin- que spatio tumidulo, latiusculo, levi, spongioso vel suberoso. Al- bumen sive Perispermium teretiusculum, intus (plano-convexum), Vitte 6, rect, wquales; 4 dorsales, quidistantes; jugis tribus dorsalibus sub-majoribus alternantes, sc. 4 intermediis opposite; relique due
* Hee forsan melius ita describenda: mericarpia jugis 5, tribus dorsalibus filifor- mibus, sub-prominulis; duobus lateralibus marginantibus dilatatis, spongiosis, spatium latum, convexum, tumidulum, suberosum utrinque formantibus ; valleculis 1-vittatis, 1-striatis ; striis filiformibus, distinctis, juga subzquantibus, sc. vix minis prominulis ; hine mericarpia 7-striata apparent. Carpophorum evanidum, sub-nullum.
of Madera and Porto Santo. 31
juncturam respicientes, approximate. Totius plant succus aquosus. An Genus?
Gren. SAMBUCUS, Linn., Spr. 52. Sambucus nigra, Linn. Sm., &e.
a. communis; foliolis ovatis. Sambucus nigra, Auct. Hab. in Europa; Anglia, &c. B. lanceolata: foliolis lanceolatis vel ellipticis, elongatis. Sambucus lanceolata, Herb. Banks. Hab. in Madere sylvis: in hortis etiam ab incolis colitur.
Preter foliola magis elongata, omnia ut in a; ideoque vix spe- cies consenda.
Orv. XV. CRASSULACEZ. Gen. SEDUM, D.C.
53. Sedum fusiforme, Prodr. MS.—Tab. 3. ff. 1, 2.
S. caule fruticuloso, ramoso; ramulis confertis, erectis, tortuosis, glabris, inferné nudis: foliis omnibus sparsis laxis sub-patentibus, car- nosis, crassis, fusiformibus, sub-teretibus, supra planiusculis, acutiusculis, utrinque attenuatis, glaberrimis, glaucis: cymis terminalibus, cormboso- fastigiatis, paucifloris: petalis 5, lanceolatis, obtusiusculis, patulis: squa- mis nectariferis brevibus, lunatis.
Hab. in Mader rupibus excelsis aridis maritimis.
Ramosissima, cespitosa, humilis. Flores flavi.
S. altissimo proxima; habitu prorsis S. nudi, cui maxime affinis. 54. Sedum farinosum, Prodr. MS.
S. candicans: caulibus herbaceis, prostratis (repentibus?), elon- gatis, inferné nudis, sub-simplicibus; foliis ad apices confertis, 4—fariis,
32 Mr. Lowe on the New Plants and Land Mollusca
caulibusque albo-farinosis, teretibus, supra sub-planulatis, obtusissimis : cymis terminalibus, 3-partitis: petalis 5, ovato-lanceolatis, acutis; squa- mis nectariferis
eee POC my eR APO) OS tach Oy Goth fy Chee UCR. he Grouch ete cc. Sify Pyne
carpellis rostratis, acutis.
Hab. in rupibus umbrosis Madera, ad altitudinem 4000 ad 5500 feré pedum.
Petala alba, nervo extra rubro.
Orv. XVI. LYTHRARIEZ. Gren. LYTHRUM, D.C. 55. Lythrum junceum, Sol. ISS.
L. floribus axillaribus, hexapetalis, dodecandris: filamentis 6 brevis- simis; 6 longioribus, tubo brevioribus; antheris sub-inclusis: calycis angulati dentibus alternis minoribus: foliis alternis, confertis, lanceolato- linearibus, sub-glaucescentibus: caulibus acuté 4~angularibus, debilibus, humifusis, elongatis; deorstim nudis, suffrutescentibus.
Lythrum junceum, Sol. MSS! et Herb. Banks. quoad specimina Maderensia !
acutangulum, Lagasca, (LL. Grefferi var? D.C.) ? Hab. in Mader humidis, frequens. Caules graciles, demiim prelongi, simpliciusculi, inferné nudi,
frutescentes. Folia parva, sub-conferta. Flores hexapetali, magnitudine mediocri, leté purpurei.
Orv. XVII ROSACESX. Gen. RUBUS, Linn. D.C. 56. Rubus grandifolius, Prodr. MS.
R. caulibus fruticosis, angulatis, aculeatis, glabris, procumbentibus, sterilibus elongatis; aculeis sparsis, compressis, recurvis, numerosis: fo- liis quinatis (rard ternatis), sub-pedatis; foliolis ovato-oblongis, acumi-
of Madera and Porto Santo. 33
natis, grossé duplicato-serratis, utrinque glaberrimis, nudis, longe peti- olulatis; petiolis petiolulisque sparsim aculeatis: panicule elongate, ter- minalis, ramis pedunculis calycibusque densé purpureo-glandulosis : laciniis calycinis reflexis, inermibus, petalis multd brevioribus.
Rubus pedatus Herb. Banks! et Sol. MSS! non Smith.
Hab. in rupibus Madere.
Folia lucida, utrinque viridia, magna. Flores albi, conspicul,
magni. Fructus sat magni, atri.
Orb. XVIII. LEGUMINOS®. Gen. VICIA, Tourn. D: C.
57. Vicia albicans, Prodr. MS.
V. annua, villosa, sub-canescens: caulibus tetragonis: cirris valde ramosis: foliolis oblongis, mucronatis, numerosis, oppositis et alternis: stipulis semi-sagittatis, inciso-dentatis: pedunculis sub-bifloris, folio mul- tum brevioribus; floribus secundis, laxis, sub-remotis: dentibus calycinis duobus superioribus minimis, obsoletis; inferioribus ovato-subulatis, medio longiore; omnibus tubo brevioribus et cum toto calyce colo- rato pilosis: stylis capitatis, infra capitulum globosum undique, subtis vero presertim, barbatis: leguminibus oblongis, latiusculis, brevibus, sub-compressis, albido-hirsutissimis, pendulis, sub-tetraspermis; semini- bus globosis, viridi-fuscis, nigro-maculatis, glabris.
Hab. in rupestribus aridis apricis Madere.
Flores magnitudine mediocri, paulld sc. majores quam in V’. Craced, rosei vel purpurei, apice purpureo-nigro, vexillo striato. V. atropur- puree, Desf: ; trichocalyci, Moris.; Broteriane, Ser. in D.C. Prodr. (V. villosa Brot non Roth.) affinis, Radice annua a V. perennt D. C.; argented, Lapeyr.; variegatd Willd. ; alpestri Stev.; cinered Bieb., necnon aliis notis distineta.
58. Vicia micrantha, Prodr. MS.
V. annua, gracilis, glabriuscula: caulibus filiformibus: cirris ra- mosis: foliolis angusto-lanceolatis vel lineari-oblongis, remotiusculis,
Vol. IV. Part I. E
34. Mr. Lowe on the New Plants and Land Mollusca
ob-tusiusculis, sub-puberulis: stipulis parvis, angustissimis, semi-sagittatis, superioribus simplicibus : pedunculis sub-bifloris, folio multtim brevio- ribus: calyce leguminibusque latis, oblongis, compressis, 3—6-spermis, villosis.
Vicia gracilis, Sol. MSS. et Herb. Banks! non Lovsl.
Hab. in Madera; Sol. e¢ Mass.
Foliola sub-octoparia. Flores perparvi, purpurascentes.
Gen. ONONIS, Linn., D. C.
59. Ononis dentata, Sol. IZS'S.—Tab. 4.
O. herbacea, annua, erecta, pilosa: foliis (omnibus) trifoliolatis ; fo- liolis obovatis, serratis: stipulis ovatis, dentatis: floribus sparsis, soli- tariis, axillaribus, pedunculatis, folio longioribus, cernuis: pedunculis muticis: corolla calycem superante: laciniis calycinis 4 supremis antice dilatatis, foliaceis (3—-) dentatis; infima simplici lineari-acuminata, in- tegerrima; leguminibus calyce longioribus.
Ononis dentata, Sol. MSS! et Herb. Banks. quoad specimina 3 in
Insulis Canariis A. D. 1778 a cl. Masson lecta!
Hab. in Portu 8S”. “Insule Canariz Fr. Masson 1778,” Herb. Banks: Yn apricis Nivarie, P.B. Webb, arm.
Flores conspicui; in plantis ab amico Rev.’ M. J. Berkeley in Anglia cultis (4 seminibus que in Insulé Portis S". mense Maii, A. D 1828, ipse legi) vexillo roseo-purpureo, alis et carina pallidioribus; in aliis (desiccatis) ab amico P. B. Webb arm. in Nivaria lectis, pallidé flavi, carina purpurea.
Gren. ASTRAGALUS, D.C. Series I]. OCHROLEUCI, §. 7. . Bucerates, D.C. Prodr.
60. Astragalus canescens, So/. MSS. A, villoso-pubescens, adscendens: caulibus diffusis, adscendentibus : foliolis multijugis, ovalibus vel oblongo-ellipticis, retusiusculis, supra
of Madera and Porto Santo. 35
glabris, infra hirtis, canescentibus: pedunculis elongatis, folio multim longioribus: racemis multifloris: pedicellis fructiferis deflexis: legu- minibus faleatis, compressis, dorso canaliculatis, apice acutis, pubescen- tibus, pendulis; sulci dorsalis lati, profundi marginibus acutis.
Astragalus canescens, Sol. MSS. et Herb. Banks!
Hab. in Insulé Portu 8S". etiam Canariis ab amico P. B. Webb arm. lectus.
Flores pallidé flavi, virescentes. 4. hamoso proxima; nec forsan vere distincta.
Orv. XIX. HYPERICINE®. Gren. HYPERICUM, D.C.
61. Hypericum angustifolium, Prodr. MS.
HT. glabrum: caulibus simplicibus erectis, strictis, virgatis, anci- pitibus, suffrutescentibus: foliis epunctatis, erectis, lineari-oblongis, \ob- tusissimis vel retusis, amplexicaulibus, margine revolutis: panicula ter- minali, corymbosa: sepalis ovatis, xqualibus, dentato-glandulosis peta- lisque nigro-punctatis: floribus trigynis.........
Hab. in Madere campo precelso (5000—6000 ped. alt.) “Paul da Serra” dicto.
Caules plures, feré pedales, tenues.
Orn. XX. MALVACE.
Gen. SIDA, Cav., D.C. Sect. MALVINDA, Med., D. C.
* * Oblongifolie ; nempe pedicellis elongatis, distinctits articulatis, foliis ob- longis ovatisve. D.C. Prodr. 62. Sida maderensis, Prodr. MS.
S. fruticulosa: foliis lanceolatis oblongisve, acutis, serratis, glabris, subtis pallidis, sub-glaucis, breviter petiolatis: axillis inermibus: pedi- cellis axillaribus, unifloris, inaqualibus, folio brevioribus: carpellis
10—12, uni-rostratis, EQ
36 Mr. Lowe on the New Plants and Land Mollusca
Malvinda unicornis folio rhomboide perennis, Déllen. Hort, Elth. p. 216, ¢. 172. f. 212. (descr. et fig. opt.)
Hab. sects vias in locis incultis &c. Madera; in regione tota in- feriore vulgatiss:
Pedicellis nunquam “ folii longitudine” et carpellis pluribus, semper uni-rostratis, & .S. canariensi differt. Flores parvi, ochracei. Fruticulus.
Osp. XXI. VIOLARIE. Gren. VIOLA, Tourn., D. C. Sect. I. Nomimiwm, Ging.—
§. 2. ——*, D. C. Prodr. I. p. 295.
63. Viola maderensis, Prodr. MS.
V. caulesceus, stolonifera: caulibus brevibus, erectis, suffrutescen- tibus, glabris: foliis profundé cordatis, rotundato-ovatis, sub-pubescen- tibus; petiolis elongatis pedunculisque pube deflexa hirsutis; stipulis glabris, acuminatis, glandulis ciliato-serrulatis: sepalis oblongis, acutis: petalis lateralibus vix sub-barbatis: caleare sub-compresso, saccato, ple- rumque obtusissimo, (rard acuminato): stigmatis rostro uncinato, de- orstum (se. ad flexuram) immarginato, nudo, complanato (nec convexi- usculo), styloque compresso, simplici, glabro: capsulis pubescentibus, hexagonis, globosis, abbreviatis: seminibus albidis, pallidé flavescentibus, obovatis.
Hab. in Madere sylvis, ubique vulgatissima.
Flores odoratissimi, violacei, sub-pallidiores quam in V.. odoratd.
Orv. XXII. CRUCIFERZ. Gren. SINAPIDENDRON, 2nob.: Prodr. MS.
Srnarpis, Sect.? 5 Disaccium, D. C Srinapis, Brown, in Hort. Kew., Hook. HespPeEris, Spr. Calyx clausus, demim erecto-patens; basi sub-bisaccatus. Stylus distinctus. Séigma capitatum. Siliqua linearis, teretius-
of Madera and Porto Santo. 37
cula, sub-torulosa, flexuoso, rostrata, basi tetragona; septo sub- spongioso. Semina uniseriata, oblonga. Cotyledones incum- bentes, sub-conduplicate.
Suffrutices Maderenses. Folia sub-carnosa, rigida, simplicia. Flores flavi, inodori. Silique graciles, elongate, pedicellate. Genus habitu.
nN
calyce, siliquis, seminibusque 4 Sinapi distinctum. SPECIES.
64. Sinapidendron frutescens, Prodr. MS.
Sinapis frutescens, Ait: Hort. Kew: IV. p.127. n°. 11.—Herb. Banks! D.C. Prodr, I. p. 220. n°. 34.—Hook: Mise. Bot. 1. p. 119. Z. 25)!
Hesperis diffusa, Spr. Syst. II. p. 900. n°. 18.
Hab. in rupibus Madere.
65. Sinapidendron salicifolium, Prodr. IS.
S. “caule frutescente; foliis lineari-lanceolatis, integerrimis.” So/. MSS.
Brassica frutescens, Sol. MLSS. et Herb. Banks!
“Hab. in Madera inter rupes maritimas prope vicum Camara de Lobos. 4. Fr. Masson.” Sol. MSS.
Species videtur a S. frutescente distincta. Folia succulenta, con- ferta, integerrima, sub-obtusa, 2—3 poll. longa, 4 poll. lata. Calya semi- patens. Siligue 1—1} poll. (absque rostro) longz, lineares, flexuosa, graciles, 4-angulares, longitudinalitér sub-striate ; rostro 4—+4 poll. longo, capitato, subulato, sub-compresso coronatz.
Habitus omnind S. frutescentis. Plantam vivam nondum vidi: descriptio 4 specimine Banksiano composita est.
66. Sinapidendron rupestre, Prodr. MS.
S. caule basi frutescente petiolis, foliisque crassiusculis, strigoso- hispidis; superioribus elongatis, oblongo-linearibus, integerrimis; infe- rioribus ovato-oblongis, sinuato-dentatis, basi sub-lyratis, petiolatis: sili- quis glabris; rostro ancipiti, brevi,
38 Mr. Lowe on the New Plants and Land Mollusca
a. chetocalyx; pedicellis, calycibus maculatis, germinibusque hispidis. Hab. in rupibus convallium Madere.
B. gymnocalyx; siliquis sub-abbreviatis, pedicellis, calycibus sub-imma- culatis, germinibusque glabris: foliis lucidis; inferioribus rotundatis obtusis, setis raris scabris.
Hab. in rupe quadam excels’ maritima, ad locum ore Septentri- onalis Madera “ Entroza” dictum: semel tantim legi.
In a, Calyx purpureo-nigro maculatus. Flores majusculi; petalo- rum limbo citrino; ungue purpureo.
8. An species? sed habitu eodem gaudet; nec in ceteris charac- teribus, floribus &e, preter supra indicata, differt.
Orv. XXIII. RANUNCULACE®, D.C.
Gren. RANUNCULUS, C. Bauh., D.C. OO: Il. Ranuneulastrum. D. C.
67. Ranunculus grandifolius, Prodr. MS.
R. foliis amplissimis, lucidis, cauleque hirsutiusculis; radicalibus petiolatis, orbiculato-reniformibus, latis, sub-quinquelobis, dentatis; lobis abbreviatis, rotundatis: caule elato, ramoso, corymboso; ramis divari- catis, sub-patentibus: corymbo vasto, amplo: calyce patentissimo.
Hab. in rupibus humidis umbrosis Madera; presertim Convallis frigid (Ribeiro Frio dictz).
Charaeteribus difficillimé, habitu se. staturé, toto ecelo ab affinibus R. cretico et R. cortusefolio dignoscitur. Plantam in horto cultam nec solo, nee ccelo aridiore mutatam inveni. Folia radicalia sub-indi- visa, diametro feré pedali. Caulis 2—3-pedalis. Flores conspicui, flavi, magni; petalis sc, 1 poll. longis.
of Madera and Porto Santo. 39
MOLLUSCA.
Crass: GASTEROPODA.
Orv. PULMONEA. 1. Familia, Limacide.
I. Genus ARION, Fer.
1. Arion empiricorum, Fer. a. Varietatis a. Fer. sub-varietates due; altera olivacea vel fusco- lutescens; altera pallidior, czruleo-cinerascens. o | Her. Hab. in Madera.
II. Genus, LIMAX. Fer. 2. Limax antiquorum. Fer. a. Fer. sub-varietates. yn. Fer.? Hab. in Madera. 8. Limax variegatus, 8. Fer. Hab. in Madera. 4. Limax agrestis, Fer. e. Fer. y. Fer. Hab. in Madera.
40 Mr. Lowe on the New Plants and Land Mollusce
III. Genus, TESTACELLUS, Cuwv. Testacellus haliotideus, Drap., Sow., Fer. Hab. in Madera. 6. Testacellus Maugei, Fer., Sow. Hab. in Madera.
ei
2. Familia, Helicide. IV. Genus, VITRINA, Drap. <
He.icouimax, Fer. 7. Vitrina Lamarckii, nob. in Zool. Journ.—Tab. 5. ff. 1, a, b.
Helicolimax Lamarckii, Fer. Hab. in Madera et Portu 8”.
V. Genus, HELIX, Fer. (excluso Sub-genere Cochlodina, i.e. Clausilia), Oss. Methodum cl. Ferussaci hine usque ad finem Cochlodontium
sequor. §§. Incluseze.
+. Volutate, Helicoides.
I. Sub-genus. HELicocrna.
1. Columellate; columella solida torta; globose.
8. Helix furva, Prodr. MS.—Tab. 5. f. 2.
H. testa imperforata, sub-globosa, tenui, fusco 1-fasciata; epider- mide umbrino: anfractibus obsoleté rugulosis, primo carinato, ceteris planiusculis: sutura distincté: spira depressiuscula, obtusa: peristomate simplici, acuto.
Axis 5 lin. Diam. 93. Anfr. 6.
a. fascia continua. B. fascia interrupta. Hab. in Madere sylvis; rarior. 9. Helix erubescens, Prodr. MS. H. testa imperforata, globosa, tenui, rubescente: anfractibus strio-
of Madera and Porto Santo. 41
lis rugisve valdé obliquis, sub-undulatis vel anastomosantibus corru- gatis; primo vix sub-carinato; ceteris convexiusculis, aquis: spira elevato-obtusa: peristomate acuto, sub-reflexo, intus sub-incrassato, carneo.
Axis 4 lin. Diam. 7. Anfr. 5. a. testa fasciis maculisve fuscis ornatéa.—Tab. 5. f. 3. 3. testa immaculata, unicolore.
Hab. in Madere sylvaticis. 10, Helix sub-plicata, Sow.—Tab. 5. f. 4.
Sow. in Zool. Journ. 1. p. 56. n’ 1. t. i. f 1! (testa decorticata, semi-fossilis.) Hab. in Insulé quadam parva, juxta Portum Sanctum, “TIlheo de Baxo” dicta. 2. Imperforate (Depresse).
Testa depressa umbilicata ; umbilico omnino tecto. Fer.
11. Helix undata, Prodr. MS—Tab. 5. f. 5.
H. testa juniore umbilicata, adulta imperforata, sub-globoso-depressa, unicolore, fusco-nigrescente : anfractibus corrugatis vel undato-rugosis, nitidiusculis; ultimo depresso, supra planiusculo; ceteris convexiusculis, sutura distincta: spira brevi, obtusa, sub-depressa: peristomate simpli- ciusculo, sub-inerassato, vix reflexo, pallido.
Axis } poll. Diam.1. Anfr. 6.
Helix corrugata, Sol. MSS.; nee Gmel. nec. Dillw.
Helix scabra, Wood's Suppl. ¢. viii. f. 62! nec Chemn., nec Lam. Feruss. &e.
Hab. in Madere sylvis, graminosis montanis, &c., vulgaris.
12. Helix phlebophora, Prodr. MS.—Tab. 5. f. 6.
H. testa juniore umbilicata, hispida; adulta imperforata, sub-glo- bosa, fusco bifasciata: anfractibus sub-tumidis, striis crebris, equalibus, transversis, obliquis, sub-flexuosis sculptis; ultimo ad angulum peristo-
matis inferiorem depressiusculo, ventricoso, prominente : spira conoidea, Vol. LV. Part I. F
42 Mr. Lowe on the New Plants and Land Mollusca
sub-exserta, obtusa ;sutura distincta, ab angulo peristomatis primo valde obliqua: apertura rotunda; peristomate continuo, simpliciusculo, paul- lum incrassato; columella expansa, plana, rosea.
Axis 4—43 lin, Diam. 8. Anfr. 54.
Helix nivosa, Sow. in Zool. Journ. 1. p. 56. n°. 3. t. iii. f 3!
Helix exalbida, Wood, Suppl. ¢. viii. f. 81!
Hab. in Insula Portis S$"; ubique vulgatissima.
Nomen alterum imponendum est ob priora (4 testis quibusdam de- corticatis, ut videtur, orta) speciei prorsus abhorrentia, ideoque difficul- tatem indagatoribus vel diligentissimis haud levem parantia. Nomen itaque novum, quodammodo aptius, ¢ duobus incommodis minus esse malum videtur: tales enim mutationes pessime auctoritatis, nec nisi gravissimis argumentis probari possunt. In dilemmate verd tali, quis inter nominis veteris plane falsi et erronei adoptionem, et aptioris quamvis
recentioris usum hzreret ?
II. Subgenus, HeLicopon. (Helicodonta Fer.)
I. Perrsonare.
Peristoma sinuatum et incrassatum, vel reflexum atque dentatum, dentibus,
laminis, plicisve tortuosis anfractiis penultimi partis convex sepe coarctatum.
13. Helix areta, Prodr. MS.—Tab. 5. f. 7.
Hi. testa rotundata, depressa, utrinque planiuscula, carinata, umbi- lico minimo perforata, solida, crassa, glabra: anfractibus striis crebris, zqualibus, transverso-obliquis crassiusculis rudibusve sculptis: spira con- vexo-depressa; sutura distincta, sub-impressa: apertura transversa, ovali, dente lamellata intis ad ventrem* coarctata; peristomate albo, reflexo, continuo, «quali.
Axis 1—1} lin. Diam. 2—21. Anfr. 4—41,
Hab. in Madere collibus aridis maritimis.
' * Venter, pars convexa anfractis penultimi, aperturam (in Helicibus) coarctans,
of Madera and Porto Santo. 43
14. Helix fausta, Prodr. MS—Tab. 5. f. 8.
HZ. testa rotundata, carinata, sub-globoso-depressa, supra* convexi- ore, pilis brevissimis undique scobinato-hispida, leviuscula: spira eleva- tiuscula, depresso-conoidea: anfractibus planiusculis, obsoletissimé trans- verse striatis; sutura distincta, impressa: apertura transversa, intis angustata, exterits dilatata, dente lamellata intts ad ventrem coarctata: peristomate extra expanso, sub-reflexo, acuto; columellam versus albo, incrassato, sub-sinuato sc. obsoleté bidentato, reflexo, umbilicum penitis obtegente.
Axis 13 lin. Diam. 3. Anfr. 54.
Hab. in sylvis Convallis “Boa Ventura” (i. e. Boni Successtis) dicte. in Maderz ora Septentrionali.
Helici personate cl: Draparnaudi maximé quidem affinis, sed
distinctissima.
15. Helix arridens, Prodr. MS.—Tab. 5. f. 9.
Hi. testa carinata, umbilico parvo perforata, rotundata, depressa, utrinque sub-planulata, tenui, hispida, leviuscula: spira convexo- depressa; anfractibus planiusculis, obsoletissimé transversé striatis; sutura sub-distineta: umbilico spiralif, rotundo: apertura edentula, transversa, ints angustata et in umbilicum quasi cum rictu paullim producta; peristomate interrupto, extra simpliciusculo, sub-reflexo; angulum_ver- sus internum incrassato, sub-sinuato, albo, reflexo, et umbilicum partim lamina expansa obtegente.
Axis 1} lin. Diam. 3. Anfr. 44—5.
Hab. in Madera.
Characteribus forté artificiosis cum Hezicert1s Hygromanibus con- socianda; sed affinitas summa cum priore reliquisque Hzzrcoponrraus
* Supra latus quo umbilicus situs est; infra quo spira, respicit. t Umbilicus spiralis dicitur ubi plus minus anfractis penultimi, antepenultimi, plu- riumye intis conspiciuntur. Huic opponitur umbilicus cylindricus.
F2
44 Mr. Lowe on the New Plants and Land Mollusca
naturalis, his adnumerare docet. Helici edentule cl: Draparnaudi proxima; caret autem impressionibus externis plicarum; caret quidem omninod plicis ipsis ullis: ideoque ab Helice faustd nostra et H. per- sonatd Drap':, aliisque hujusce sectionis cognatis, nullo modo intervallo longo separanda est.
III. Subgenus, Hrevicicona. 1. Carocolle. Uwmbilicus tectus. 16. Helix Webbiana, Prodr. MS\—Tab. 5. ff. 10.
HZ, testa adulta imperforata, tenui, nitida, sub-lampadiformi, de- pressa, carinata, utrinque convexa, corneo-fuscescente; supra ad umbi- licum virescente, convexiore, oblique tenuiter striata, carinam versus utrinque sub-impressam, obtusam, suturamve granulis minutis scabra: spira convexo-depressa, obtusissima; sutura distincta; anfractibus pla- nulatis, ultimo maximo: apertura transversa, sub-ovali, amplissima, patula, extra carina angulata; peristomate interrupto, tenui, acuto; extra valde expanso, patulo; ad columellam sub-incrassato, sub-reflexo.
Axis 3lin. Diam. 9. Anfr. 3—34.
Hab. in montibus Insulz Portis 8".
Amico P. B. Webb, Arm’., Natur indagatori impigro ac _peritis- simo, speciem pulcherrimam atque rarissimam dico.
2. Vortices. Umbilicus apertus, 17. Helix Bulveriana, Prodr. MiS_—Tab. 5. ff. 11.
HZ, testa rotundato-depressa, hemispherica, rotata, supra planulata, acutissimé carinata, tenui, nitidiuscula, tota minutissimé et confertim granulata, fusco-castanea, supra fasciata: spira convexo-depressa, plus minus elevata, obtusissima; sutura obsoleta; anfractibus planis, xquis, quasi attritis vel confluentibus, ultimi cariné acutissima, tenui, supra sulco exarata, limbata; umbilico patulo, spirali, profundo: apertura
rotundato-lunata: peristomate interrupto, ad umbilicum incrassato, re- flexo.
of Madera and Porto Santo. 45
Axis 3—21 lin. Diam. 7—8. Anfr. 8—7.
Helix Bulverii, Wood, Suppl. t. vii. f- 82!
Hab. in montibus Insule Portis $8". 18. Helix tectiformis, Sow —_Tab. 5. f. 12.
Sow. in Zool. Journ. p. 57. n’. 6. t. it. f 6!
Hab. in insula quadam parvulé “Theo de Baxo” dicta juxta In- sulam Portis 8".
19. Helix subtilis, Prodr. MS.—Yab. 5. f. 13.
H. testa orbiculari, utrinque depresso-planulata, tenui, unicolore, pallidé fusco, acuté carinata: spira depressa, sub-planulata; sutura dis- tincta; anfractibus planulatis, striis transversis, obliquis, tenuibus, plus minus distinctis, equidistantibus, interstitiisque striolis subtilissimis, crebris, decussatis: umbilico patulo, magno: apertura transversa, de- pressa, obliqué lunata; peristomate interrupto, sub-simplici, sub-reflexo.
Axis 1 lin, Diam. 3—4. Anfr. 5.
An Helix lenticula, Feruss. Tabl. Syst, n°. 154?
Hab. in Maderee maritimis.
20, Helix actinophora, Prodr. MS.—Tab. 5. f. 14.
H. testa orbiculata, depressa, supra convexiore sub-turgida, tenul, unicolore, fusco-rufescente, acuté carinata: spira convexiusculo-depressa, sub-planulata; sutura distincta; anfractibus planatis, striis creberrimis, tenuissimis, transversis undulatim laminosis, quibusdam ad carinam su- turamve in laminas breves, membranaceas, lacinulasve acutas, radiantes productas, notatis: umbilico spirali, parvo: apertura transversa, rotun- dato-ovali, sub-lunata; peristomate interrupto, acuto, patulo, reflexo,
Axis 2 lin, Diam. 4. <Anfr. 5.
Hab. in Madere sylvaticis.
IV. Subgenus, HELIcELLA.
Lomastome ; peristoma reflexum,
21. Helix pulchella, Mudd. Hab. in Madera,
46 Mr. Lowe on the New Plants and Land Mollusca
292. Helix Porto-sanctana, Sow: a. vulgaris.—Tab. 5. f 15. Sow: in Zool. Journ. 1. p. 57. n’. 5. t. iit. ff. 5! Hab. copiosissimé in Portu 8”, B?2 gigantea—Tab. 5. f. 16.
Hab. in Portu 8”.
An var. 8. species potitis? quamvis enim ab a. vix nisi magni- tudine duplo feré majore differt, status intermedios nunquam vidi Var. a. viva ubique copiosissima; #. rarissima nondum nisi statu semi- fossili, decorticato occurrit.
Aplostome ; peristoma simplex. * Verticilli. 23. Helix pusilla, Prodr. MS—Tab. 5. f. 17.
HZ. testa rotundato-depressa, ecarinata, tenui, rufescente: spira con- vexiuscula; sutura distincta, impressa; anfractibus rotundatis, _ striis transversis, annularibus, elevatis, sub-membranaceis, tenuibus, remotis, equalibus, plicatis; interstitiis striolis aliis exilissimis tenuissimisque creberrimis, spiralibus sc. transversas decussantibus, sculptis: umbilico patulo, spirali, profundo: apertura rotunda, vix lunata se. circuli seg- mento perparvo dempto; peristomate simplici (tenui, acuto).
Axis $ lin. Diam, 1. Anfr. 4.
Hab. in Madere sylvis.
Obs. Helici pygmee cl: Drap: quoad staturam et habitum maximeé affinis; species autem revera distinctissima.
* * Hyaline. 94. Helix bifrons, Prodr. MS —Tab. 5. f. 18.
H. testa rotundato-depressa, umbilicata, sub-carinata, tenui, nitida, concolore, corneo-virescente; supra leviuscula, obsoleté striata; inferneé striis valde distinctis sculpta: spira convexo-depressa; sutura distincta, impressa; anfractu ultimo infra carinam, ceterisque anfractibus, striis costisve transversis, zqualibus, crebris sulcatis: umbilico parvo, cylin-
of Madera and Porto Santo. 47
drico, sub-spirali, profundo: apertura lunata; peristomate simplici, tenui, acuto, intus albo marginato.
Axis 23—3 lin. Diam. 6—7. Anfr, 7—8,
Hab. in Maderz sylvis, 25. Helix cellaria, Mull.
Helix lucida, Mont.
Hellx nitens, Drap.
Hab. in Madera.
26. Helix erystallina, Mull, Hab. in, Madera.
Heliomanes ; peristoma marginatum (“ bordé”).
* Testa depressa vel globulosa.
27. Helix paupercula, Prodr. MS.—Tab. 5. f. 19.
HZ, testa rotundata, planata, supra conyexiore, umbilicata, sub-ca- rinata, solidiuscula, rudi, feré unicolore, anfractu ultimo supra carinam obsolete fusco unifasciata: spira planata; sutura impressa; anfractibus rugosis, eroso-scrobiculatis, minutissime elegantissiméque granulatis; ul- timo ad aperturam constricto: umbilico largo, patulo, spirali, profundo: apertura rotundata, coarctata; peristomate continuo, eley ato-disjuncto, annulari, sub-patulo, acuto; labro intts 1—dentato.
Axis 1 lin. Diam. 2—2t, Anfr, 344,
Hab, in Maderz et Portis S*. maritimis.
28. Helix obtecta, Prodr. MS.—Tab. 5. ff. 20, a, b.
HI. testa rotundata, depressa, inferné planata, supra convexa, um- bilicata, carinata, solidiuscula, rudi, albida, limo vel terra obducta: spire planulate anfractibus primis concavyis, ceteris prominentibus tur- gidis; sutura distincta, valde impressa; anfractu ultimo ventricoso, carina distincta, utrinque sub-exarata vel sulco obsoletissimo expressa ; omnibus rugosis, eroso-scrobiculatis, minutissimé elegantissiméque gra-
48 Mr. Lowe on the New Plants and Land Mollusca
nulatis: umbilico mediocri, sub-spirali: apertura rotundata; peristomate continuo, sub-disjuncto, tenui, acuto, sub-expanso, intus incrassato.
Axis 2 lin. Diam. 5. <Anfr, 4}—5.
Hab. in montibus collibusque aridis Portis S". rarior; copiosior in Insula “Theo de Baxo” dicta.
Przcedentis forsan status, vel varietas tanttim major.
29. Helix dealbata, Prodr. MS.
H. testa rotundato-depressa, utrinque convexa, umbilicata, carinata, solidiuscula, albida: spira convexo-depressa, sub-conoidea; sutura sub- distincta; anfractibus sub-planulatis, transversé rugoso-striatis, plerum- que minute granulatis; ultimo obtuse carinato: umbilico parvo, patulo, sub-spirali, minime profundo; apertura rotunda, ochracea; peristomate continuo, reflexo.
Axis 2lin. Diam. 4—44, altera transversa 3—3}. Anfr. 6.
a. granulata; testa granulata—Tab. 5. f. 21.
Hab. in montibus Portis 8".
B. levis; testa egranulata, levi, nitida.
Hab. in insula “Tlheo de Baxo” dicta. Status a, solo calcareo ortus.
30. Helix maderensis, Wood.—Tab. 5. f. 22.
H. testa rotundato-depressa, utrinque planulata, umbilicata, cari- nata, solidiuscula; supra leviuscula, fusco 1-fasciata; inferné striata: spira convexiuscula, sub-planulata; sutura distincta; anfractibus pla- natis, infra transverse striatis, ultimi ad aperturam granulati carina acuta, striis supra carinam obsoletis: umbilico lato, patulo, spirali:
apertura rotunda; peristomate continuo, circinato, annulati sub-dis- juncto, crassiusculo, sub-expanso.
Axis 14 lin., rariss. 2; Diam. 3, rariss. 4. Anfr. 6—7. Helix maderensis, Wood. Suppl. t. viii. f. 84! Hab. in Madera; vulgatissima.
31. Helix compar, Prodr. MS.—Tab. 5. f. 23.
HI. testa rotundato-depressa, utrinque planulata, umbilicata, sub- carinata, solidiuscula, fusco bifasciata, utrinque plicato-costata: spira
of Madera and Porto Santo. 49
conyexiuscula, sub-planulata; sutura distincta, impressa; anfractibus convexiusculis, plicis vel striis transversis, elevatis, acutis, distinctis, crebris, zquidistantibus aqualibusque costatis; interstitiis levibus; ul- timi carina obtusa: umbilico lato, patulo, spirali, profundo: apertura rotundato-oyali; peristomate continuo, circinato, sub-disjuncto, crassius- culo, reflexo.
Axis 13 lin. Diam. 34. Anfr. 6.
Hab. in Madere collibus maritimis; rariss.
32. Helix leptosticta, Prodr. MS—Tab. 5. f. 24.
HZ. testa rotundato-depressa, umbilicata, sub-carinata, nitidiuscula, tenui, pallidé cornea, obsolete fasciata: spira convexo-depressa; sutura distincta; anfractibus convexis, sub-striatis, minuté et elegantissimé re- ticulato-granulatis ; ultimi carina obtusa: umbilico patulo, spirali: aper- tura rotundato-ovali; peristomate continuo, simpliciusculo, sub-incras- sato, sub-reflexo.
Axis 1} lin. Diam. 3. Anfr. 5—51.
Hab. in Mader collibus maritimis.
33. Helix lentiginosa, Prodr. MS'—Tab. 5. f. 25.
H. testa rotundato-depressa, supra sub-planulata, umbilicata, sub- carinata, tenui, maculata et sub-fasciata: spira convexo-depressa ; sutura distincta; anfractibus convexiusculis, striato-scobinatis vel squamuloso- cancellatis, striis se. interruptis squamiformibus, lunatis, quincuncialibus sculptis: umbilico mediocri, sub-patulo, spirali: apertura transverse ovali, sub-lunata ; peristomate interrupto, reflexo.
Axis 14 lin. Diam. 2}—3. Anffy. 5.
Hab. in Madere rupibus maritimis.
Helici arridenti nob: affinis. 34. Helix calva, Prodr. MS.—Tab. 5. f. 26.
H.. testa rotundato-globulosa, sub-depressa, imperforata, vix sub- carinata, nitidiuscula, sub-tenui, sub-pellucida, obsoletissimé 2—fasciata :
spira convexa, elevatiuscula; sutura distincta; anfractibus planiusculis, Vol. lV. Part I. G
50 Mr. Lowe on the New Plants and Land Mollusca
transverse costulato-striatis, striolisque spiralibus _obsoletissimis, sub- tilissimis exilissimisque, «quis notatis; ultimo feré ecarinato, ad aper- turam ochraceo, supra nitido, levi: umbilico clauso: apertura trans- versa, multd latiore quam alta, sub-lunata, ints angustata, extrorsum ampliore; peristomate longé interrupto, incrassato, sub-reflexo.
Axis 2—21 lin. Diam. 34—4. Anfr. 63—7.
Hab. in Madere sylvis.
Helici edentule Drap*. aliquatenis forma affinis; sed vix HELIco- poNTIBuUs releganda.
Hewiceiiis Hyalinis affinitate naturali, et pra ceteris H. bifronti nostre accedit; peristomate verd marginato, striisque subtilissimis te- nuissimisque spiralibus prorsts aliena.
35. Helix abjecta, Prodr. MS—Tab. 6. f. 1.
H. testa parvula, rotundato-pyramidata, conoidea, carinata, umbili- cata, crassa, solida, rudiuscula, utrinque scabra vel granulosa, rugosa, supra carinam fusco pallidé unifasciata: spira convexa, conoidea; sutura distincta; anfractibus compactis, convexiusculis, transversé rugosis et granulatis: ultimi carina sub-acuta, ad suturam approximata: umbilico parvo, spirali, profundo: apertura rotundata; peristomate continuo, reflexo.
Axis 13—2 lin. Diam. 3—3}. Anfr. 64—7.
Hab. in insula Portas S"., una cum H. compacta degens; vulga- tissima.
Inter H. compactam et H. echinulatam nostram quasi intermedia ; ab utraque satis distincta: priori verd quam maximé affinis. .
36. Helix compacta, Prodr. MS.—Tab. 6. f. 2.
H. testa parvula, rotundato-globulosa, sub-conoidea, perforata, sub- carinata, crassa, solida, rudiuscula; infra’ scabra, rugosa; supra leviore, nitidiuscula, pallidiore, fusco obsoleté 1-fasciata: spira convexa, eleva- tiuscula; sutura distincta; anfractibus compactis, planiusculis, transverse striatis et granulatis; ultimo sub-carinato, supra angulum leviusculo
of Madera and Porto Santo. 51
se. egranulato, umbilicum coarctante: umbilico minimo, sub-spirali, rimaformi: apertura rotundato-lunata; peristomate interrupto, (labris approximatis, aliquando sub-continuis) sub-reflexo,
Axis 2 lin. Diam. 3. Anfr. 6—64. Sowerb. in Zool. Journ. I. ¢. iii. f. 8!
Hab. in Insula Portis S". gregaria, ubique copiosissima: in Ma- dera ad Promontorium S". Laurentii (“Ponta Sad Lourenco”) soliim.
37. Helix consors, Prodr. MS.—Tab. 6. f. 3.
H. testa rotundato-depressa, perforata, vix sub-carinata, crassius- cula, solida, rudiuscula; infra preesertim scabra, rugosa; supra leviore; utrinque pallido fuscoque variata, ad aperturum ochracea: spira con- vexo-depressa; sutura sub-indistincta; anfractibus planatis, transverse striatis et granulatis; ultimo sub-carinato, umbilicum coarctante, gra- nulis supra angulum obsoletis: umbilico minimo, sub-spirali, rimzformi : apertura rotundato-lunata; peristomate distincté interrupto, sub-reflexo.
Axis 24—3 lin. Diam. 44—5. Anfr. 6—61. Hab. in Insula Portas S". cum precedente; rarior.
Precedenti vel maximé affinis; characteres itaque extricatu diffi- cillime: sed forma magis depressa numerusque anfractuum isdem, quamvis testa feré duplo major, speciem esse distinctam suadent; ob- stante nulla differentia loci, soli, cibi, nee alia quapiam hujusmodi causé que talem mutationem efficere posset.
38. Helix depauperata, Prodr. MS.—Tab. 6. f. 4.
H. testa rotundato-depressa, umbilicata, ecarinata, tenuiuscula, su- pra convexa, unicolore, sordida, minutissimé et elegantissimé confertim reticulato-granulata : spira convexo-depressa; sutura distincta, sub- impressa; anfractibus convexis, sub-tumidulis, transversé sub-striatis ; ultimo sub-rotundato: umbilico mediocri, aperto, spirali, profundo:
G2
52 Mr. Lowe on the New Plants and Land Mollusca
apertura rotundata; peristomate sub-continuo, simpliciusculo, tenui, intus sub-marginato.
Axis 2—21 lin. Diam. 4-43. Anfr. 5—5}.
Hab. in montibus Insule Portis S$".
39. Helix lurida, Prodr. MS.—Tab. 6. f. 5.
HZ. testa sub-globosa, depressiuscula, supra convexa, umbilicata, ecarinata, tenuiuscula, fusco sub-fasciata, nitidiuscula: spira conyexo- depressa; sutura distincta; anfractibus convexis, minutissimé et ob- soletissimé confertim reticulato-granulatis; ultimo rotundato, juxta suturam granulato, superné levi sc. egranulato: umbilico parvo, cylin- drico, profundo, aperto: apertura lunata, sub-ovali; peristomate sim- plici.
Axis 3 lin. Diam. 5. Anfr. 54—6.
Hab. in montibus Insule Portis S*.
Sequenti proxima. 40. Helix nitidiuscula, Sow—Tab. 6. f. 6. Sow. in Zool. Journ. I. Poin, Ac bil. fey Hab. in Madera et Portu S°; ubique vulgatissima. 41. Helix punctulata, Prodr. MS.—Tab. 6. ff. 7, 8. . setulosa; testa sub-tenui, sub-inflata, scabra, spinelloso-hispida. Axis 4 poll. Diam, 3. Anfr. 5.—Tab. 6. f. 7. Helix punctulata, Sow. in Zool. Journ. 1. p. 56. n°. 2. ¢. iti. f. 2! B. solida; testa solida, glabriuscula, pallida. Axis 3 poll. Diam. 3. Anfr. 5.—Tab. 6. f. 8. Hab. in Portu 8», An satis ab. H. nitidiusculd Sow. distincta ? 42. Helix pisana, Mull. H. rhodostoma Drap., cmgenda Mont., &c.
Rg
of Madera and Porto Santa. 53
Hab. in Mader Promontorio “P". Sad Lourenco” dicto. In
Portu 8S”. vinetarum calamitas.
43. Helix lauta, Prodr. MS\—Tab. 6. f. 9.
H. testa sub-globosa, supra convexa, umbilicata, ecarinata, tenui- uscula, (alba, fasciis angustis, interruptis, fuscis, obsoletis ornata), niti- diuscula: spira convexo-depressa, sub-elevata; sutura distincta; anfrac- tibus convexis, striis confertissimis, equalibus, concinnis, transversis sculptis; ultimo rotundato: umbilico parvo, cylindrico, profundo, aperto: apertura lunata, sub-rotunda; peristomate acuto, intts annulo distincto, elevato, margini approximato.
Axis } poll. Diam: %. Anfr. 5.
Hab. in Portu 8".
Specimen unicum decorticatum tantim habeo, a Rev’. Dom’.
Bulwer repertum, quod mihi cl. G. B. Sowerby humanissimé commu- nicavit. H. luride nostre, necnon H. striate Drap: (caperate Mont :) et forsan aliis quibusdam proxima: sed ab omnibus distincta videtur.
44, Helix striata, Drap*.?
Hab. in Madera; rariss.
Differt umbilico et numero anfractuum, pro magnitudine, majore. Quum autem testas perpaucas easque nondum adultas adhuc repertas
habeo, distinguere vix audeo.
45. Helix rotula, Prodr. MS.—Tab. 6. f. 10.
Hi. testa rotundata, conoideo-depressa, supra sub-planulata, sub- perforata, carinata, scabra, nitidiuscula, fasciata: spira conoidea, obtu- sissima; sutura obsoleta; anfractibus planis, transverse striatis et granu- latis; ultimo acute carinato, carina ad peristoma obsoleta: apertura lunata, extrorsum dilatata; peristomate intis incrassato, acuto, sub-ex- panso; ad angulum internum reflexo, calloso, perforationem obtegente.
Axis 3 lin. Diam. 6. Anfr. 8
Hab. in montibus Porttis S*.
54 Mr. Lowe on the New Plants and Land Mollusca
46. Helix polymorpha, Prodr. MS.—Tab. 6. ff. 11—16.
HZ testa rotundato-depressiuscula, umbilicata, carinata, crassiuscula, solida, fusco fasciata et maculata: spira conoideo-depressa, aliquando feré planata, granulata; anfractibus planiusculis; primorum saltem su- tura obsoleta; ultimi carina plis minis acuta: umbilico patulo, spi- rali, largiusculo: apertura lunato-rotundata; peristomate sub-reflexo.
a. irrasa; testa depresso-conoidea, sub-globulosa, utrinque granulato- scaberrima, limo vel terra obducta: spira convexo-elevatiuscula, conoidea; anfractibus convexis; sutura distincta; carina obtusa: peristomate sub-interrupto.
Axis 3 lin. Diam. 5. Anfr. 8.
Albida, fasciis fuscis distinctis, superiore lato, continuo, distinc- tissimo; infra sc. spira sub-maculata, variegata.
Pn ive
Hab. in solo rubro “Tufa” Geologicis dicto, ad promontoriam St, Laurentii Madere.
B. depressiuscula; testa rotundato-depressiuscula, obsoleté utrinque gra- nulata, supra presertim nitidiuscula, leviuscula sc. granulis raris, obsoletis: spira convexo-depressiuscula; anfractibus convexiusculis ; sutura distincta; carina obtusa: peristomate interrupto.
Axis 24 lin. Diam. 5—53. Anfr. 7.
Supra albida, fasciis fuscis, superiore lato, continuo, distincto, ce- teris interruptis vel obsoletis; infra sc. spira albido fuscoque maculata, variegata.
ey IED res
Hab. in solo Tufa dicto in collibus maritimis prope urbem Fun- chalensem Madere.
y- arenicola; testa rotundata, supra sub-planulata, utrinque granulata ; supra presertim nitida, granulis obsoletioribus: spira convexo-
of Madera and Porto Santo. 55
depressiuscula, plis minus elevata; anfractibus convexiusculis; su- tura distincta; carina sub-obtusa: peristomate sub-continuo.
Axis 2—21 lin. Diam. 44—5. Anfr. 7.
Sub-var. 1. Supra fusco fasciata; fascia superiore distincta, sub-continua, angusta; ceteris interruptis.
f. 13, l. ¢. 2. Inornata se. non fasciata, variegata. Albida, nitida, spira fusco maculata, variegata. Colores quodam- modo letiores quam in ceteris; albo prasertim clariore. Status a, é solo ealcareo ortus.
Hab. in arenosis calcareis Promontorii S*. Laurentii Madere.
8. attrita; testa rotundato-depressa, rotata; infra planulata; supra con- vexa, nitidiuscula; utrinque confertim granulata: spira convexo- planata; anfractibus planis, quasi attritis; sutura obsoletissima ; carina acutissima: umbilici margine (presertim in junioribus) abrupto, declivi: peristomate feré interrupto.
Axis 2 lin. Diam. 44—51. Anfr. 7. Sub-var. 1. Supra pallida, fusco fasciata; fascia superiore angusta; ple- rumque unica. f. 14, l. ec. Helix tectiformis, Wood. Suppl. t. viii. f. 83!
2. Tota fusca, sub-unicolor, preter spatium vel fasciam latam pallidam cirea umbilicum.
3. Tota variegata, nec fasciata. Sub-varietas quaque colore magis fusco quam in ceteris gaudet:
in 1™ sordidé albido vel pallidé ochraceo fuscoque variegata et macu-
lata; anfractis ultimi pars semper in omnibus juxta peristoma ochracea, immaculata.
Hab. in collibus montibusve Portis S".
56 Mr. Lowe on the New Plants and Land Mollusca
«. calcigena; testa rotundato-depressa; supra planulata, levi, nitida, ad aperturam tantiim sub-granulata: spira convexo-depressa, pltis mi- nus elevata, granulata; anfractibus planatis; ultimi sutura im- pressa, ceterorum obsoleta; carina sub-acuta: peristomate sub- interrupto.
Axis 2—21 lin. Diam. 5—5$. Anfr. 73. Sub-var. 1. Supra tota alba; spira albida fusco variegata. ii Tish lero! 2. Supra fasciata; spira albida fusco variegata. Status 4. vel ¢., solo calcareo ortus. Hab. in solo calcareo Insule cujusdam, ‘“ Baxo” dicta, juxta Por-
tum S™”,
¢. pulvinata; testa rotundato-conoidea, utrinque confertim granulata : spira elevata, conica, anfractui ultimo quasi superimposita; anfrac- tibus (preter primos) convexiusculis, ultimo tumidulo; sutura dis- tincta, impressa; carina sub-obtusa: peristomate continuo.
Axis 21—3 lin. Diam. 5. Anfr. 73.
Sub-var. 1. Supra tota alba spira sub-maculata. 2. Supra fusco fasciata; spira maculata—t. 16, l. c.
Colores in utroque statu (sc. sub-varietate) quam in ceteris varie- tatibus longé pallidiores. Testa quidem in omnibus pallida, albida, apice spire fusco.
Hab. in montibus collibusve Portis S“.; cum 36. aétrita nostra de- gens.
Varietates 6. et ¢ (forsan etiam e) primo aspectu distinctissime, tot forsan species constituende quibusdam videantur. Aded tamen, medi- ante ¢, sunt conjunctae, ut tres ille 4, «, ¢ nee a seipsis nec ab a, B, y, quibus ordine inverso analog sunt, separari debent. Sed in re tam dubia, non is sum qui cuilibet meas varietates pro speciebus habenti, increparem.
* * Testa trochoidea, carinata.
of Madera and Porto Santo. 57 47. Helix cheiranthicola, Prodr. MS.—Tab. 6. f. 17.
HI. testa pyramidata, conoidea, umbilicata, carinata, solidiuscula, tota scabra, plerumque fasciata: spira elevata, pyramidata, obtusa; su- tura distinctissima, impressa; anfractibus convexis, tumidis, distinctis, confertim granulatis; ultimi carina obtusa: umbilico mediocri, patulo, spirali profundo: apertura rotundata; peristomate continuo, sub-dis- juncto se. circinato, incrassato, sub-reflexo.
Axis 3 lin. Diam. 4. Anfr. 8.
Sub-var. 1. xonata; supra fasciata: spira fascia, unica, lata, juxta sutu- ram: carina albida.
ie Jeri 1G.
2. maculata; supra fasciata: spira maculata vel variegata, nec fasciata. 3. albida; tota albida, nee fasciata: spira sub-maculata.
Hab. in arbusculis Cheranthi tenuifolii Herit: in monte Portis S". quodam “ Pico branco” dicto: et in Insula “Ilheo de Baxo” dicto, sed rarissima.
Varietati ¢. pulvinate Helicis polymorphe nimis forsan affinis: sed forma et anfractibus tumidis et sutura impressa dignoscitur.
48. Helix oxytropis, Prodr. MS.—Tab. 6. f. 18.
H. testa depresso-conoidea, supra planulata, perforata, carinata, tota scabra, fusca, sub-fasciata: ‘spira depresso-conica; sutura distincta ; anfractibus planiusculis; ultimi carina acuta, distinctissima, supra mar- ginata sc. exarata vel suleo expressa; omnibus distinctissimé confertim granulatis, asperis: umbilico minimo, sub-spirali, aperto: apertura ro- tundata; peristomate continuo, circinato, disjuncto, reflexo.
Axis 21 lin. Diam. 4. Anfr. 64.
Hab. in collibus maritimis Portis S*.
49. Helix echinulata, Prodr. MS'—Tab. 6. f. 19. H. testa parvula, conoidea, sub-pyramidata, depressiuscula, supra
planulata, perforata, carinata, tota scaberrima, fusca, supra fasciata: Vol. IV. Part. I. H
58 Mr. Lowe on the New Plants and Land Mollusca
spira pyramidata elevata; sutura distincta, impressa; anfractibus con- vexis; ultimi carina acuta, distincta, supra marginata sc. sulco ex- pressa vel exarata; omnibus granulis distinctissimis, confertis, asperri- mis scobinatis et quasi echinulatis: umbilico parvo, sub-spirali, aperto; apertura rotundata; peristomate continuo, circinato, disjuncto, reflexo.
Axis 2 lin. Diam. 2}. Anfr. 6.
Hab. in monte “ Pico branco” dicto Insule Portis S".
Species elegantissima.
50. Helix duplicata, Prodr. MS.—Tab. 6. f. 20.
Helix bicarinata, Sow. in Zool. Journ. 1. p. 58. n°. 7. t. iti. f. 7! Wood, Suppl. t. viii. f. 85! non Feruss.
Monstrosa; anfractu ultimo disjuncto; sutura profunda, excavata.
Hab. in Insulé Portis S“.
Nomen egré, et quasi coactus, mutavi; ob Helicem C. bicarina- tam cl: Ferrussaci, Tabl. Syst. n°. 350.
51. Helix turricula, Prodr. MiS.—Tab. 6. f. 21.
Hi. testa turrita, pyramidata, sub-cylindrica, bicarinata, perforata, tota minuté et confertissimé granulata, fusca, feré unicolore, vel supra obsoleté fasciata: spira valde elevata, obtusissima; sutura distincta ; anfractibus bicarinatis, carinis equalibus, prominentibus, distinctis, sulco divisis: apertura rotunda; peristomate continuo, circinato, disjuncto, tenui, reflexo.
Axis 4 lin. Diam. 3. Anfr. 8—84.
Hab. in Insulé quadam “Tlheo de Cima” dicta, juxta Insulam Portum S™™.
Species notabilior, elegans. 52. Helix bicolor, Prodr. MS.—Tab. 6. f. 22.
H. testa globuloso-conoidea, sub-imperforata, leviuscula, nitida, vix sub-carinata, fasciis albis fuscisque lete-coloribus ornata: spira_elevati- uscula, obtusissima; sutura distincta; anfractibus sub-planulatis, trans-
of Madera and Porto Santo. 59
versé striatis: apertura extrorsim ampliore; peristomate longé inter- rupto, tenui, simpliciusculo, inttis ad angulum incrassato, reflexo, per- forationem minimam feré obtegente.
Axis 2 lin. Diam. 3. Anfr. 7.
Hab. in summo cacumine montis “ Pico de Facho” dicto Porttis S“.
Species nitidissima, coloribus distinctissimis sc. fasciis laeté coloratis gaudens. Ob affinitatem Helict maritime Drap., cel: Ferussaco obse- cutus, hue relegavi; sed ambe potits priori sectioni post Helicem
variabilem Drap. (H. virgatam, Mont.) inserende sunt.
+ Evolutate, Cochloides. * Apertura feré edentula.
1. Columella solida. ——, planata et ad basin _truncata.
5. Sub-genus, Cochlicopa.
Styloides; testa turrita, apertura brevi, &e.
53. Helix C. acicula, Fer.
Bueccinum Acicula, Mul/—Buce. terrestre, Mont.—Bulimus Acicula. Brug. et Drap.—Achatina Acicula, Lam* et Nils. Hab. in Madera.
54. Helix C. tornatellina, Prodr. MS.—Tab. 6. f. 23.
HZ. testa obovato-oblonga vel obconico-cylindrica, levi, nitida, cor- neo-rufescente (castanea): spira breviuscula, obtusa, duas partes ex quinque totius longitudinis wquante; anfractibus planis; sutura obso- leta: apertura longitudinali, coarctata, postice valde angustata; labro anticé producto, porrecto, sub-inflexo, posticé sub-sinuato: columella prominula, abrupté et oblique truncata, torta; plica in ventrem lon- gitudinali, sub-obsoleta, callosa, labro adversa, ad partem posticam an- gustatam aperture, hane coarctante.
H2
60 Mr. Lowe on the New Plants and Land Mollusca
Long. 4—5 lin. Diam. 2—27. Anfr. 7. Spira Apertura’ Hab. in Madera. Hee et 3 forsan sequentes Helici folliculo affines. 55. Helix C. melampoides, Prodr. MiS.—Tab. 6. f. 24.
TE. atest; Mbovate-oplangay eae. es, LORE SS
2 3°
2 3
wae ee wal eae 20s 25000). 2 spira” breviuseula, obtusissima, partes duas fereé ex quinque totius longitudinis zquante; anfractibus planis, ultimo sub-ventricoso; sutura obsoleta: apertura longitudinali, anticé effusa, sub-patula, omnind edentula; labro recto, equali: columella ob- soleta, obliqué truncata.
Long. 54 lin. Diam. 2}. Anfr. 6.
Spiral 4 Apertura’ *
Hab. in Insulé quadam, Portum Sanctum ab oriente spectante,. “Tlheo de Cima” dicta. v. m.
Priori nimis affinis, et forsan varietas tanttm; at major, aliquan- tulum feré ventricosior, apertura semper edentula, anticé magis effusa, posticé minus angustata, columella obsoletiore, et labro recto, quali, nec sinuato, nec anticé producto. Testa decorticata, crassa, solida, opaca; sed hee etiam in priore (H. éornatellina), post mortem animalis obtinent: vivam nondum vidi.
56. Helix C. tritieea, Prodr. MS.—Tab. 6. ff. 25, 26.
H. testa obovato-cylindrica, sub-gracili, sub-conica, nitida, levi: spira acutiuscula, dimidium teste aquante; anfractibus planis; sutura obsoletiuscula: apertura obovata, biplicata; plica altera transversa, inter columellam. et angulum labri in medio posita, altera magis interna mi- nore in columellam; duabus aliquando obsoletis; columella anticé lata,
sub-expansa, plana, vix truncata, in labrum simplex rectum equale atte- nuata.
\
of Madera and Porto Santo. 61
Long. 3 lin. Diam. 14. Anfr. 6. a. biplicata; apertura 2—plicata—f. 25, 1. ¢. B. edentula; plicis obsoletis.—f. 26, 1. e. Hab. in Portu S*.
57. Helix C. ovuliformis, Prodr. MS.—Tab. 6. f. 27.
H. testa angusto-elliptica, sub-pupeformi, diametro utrinque zquali, abbreviata, nitida, levi: spira obtusissima, dimidium teste aequante ; anfractibus convexiusculis, sub-tumidis; sutura distincta: apertura ob- ovata, angusta, biplicata; plica altera transversa, abrupta, prominente, inter columellam et angulum labri in medio posita; altera ad colu- mellam, magis obsoleta, obliqua: columella expansa, tenui, torta, obli- que truncata.
Long. 2 lin. Diam. 1. Anfr. 4,
Hab. in cacumine montis “Pico de Facho” in Insula Portis S*.
58. Helix C. gracilis, Prodr. MS.—Tab. 6. f. 28.
HI. testa elongato-obovata, gracili, tenui, vitrea, nitida, levi, (im- perforata): spira sub-attenuata, obtusa, dimidium teste excedente; an- fractibus planiusculis; sutura obsoletiuscula: apertura obovata, eden- tula: columella lata, expansa, vix truncata, in labrum tenue, sub-mar- ginatum attenuata.
Long. 2 lin. Diam. 1. Anfy. 5,
Hab. in monte “Pico Branco” Insule Portis St“.
Facies Helicis (Cochlicelle) Clavuli Fer. (H. Goodalli, Mill. An- nals of Philos.); sed magis turrita; anfractu ultimo cum penultimo majore; sutura obsoletiuscula, non distincta, impressa; testa lavissima, imperforata, nec striata, nec sub-perforata. Inter Helicem triticeam et Helicis lubrice varietatem nostram quodammodo media, ab utraque distincta.
59. Helix C. lubrica, Mudl.—Tab. 6. f. 29.
Var. testa aperturaque angustiore, minus ventricosa, magis elongata. Hab. in Madera.
62 Mr. Lowe on the New Plants and Land Mollusca
re)
Testa perforata vel umbilicata &e.; peristomate simplici. a. Anfractibus zqualibus, ultimo ceteris omnibus breviore.
6. Sub-genus, Cochlicella. 60. Helix C. ventrosa, Fer. Bulimus ventricosus, Drap. Hab. in Madera et Portu 8”. 61. Helix C. decollata, Linn. Bulimus decollatus, Drap. Hab. in Madera.
* * Apertura feré dentata vel laminata. 1. Ecanaliculate ; peristomate plerumque non continuo.
7. Sub-genus, Cochlodon. (Cochlodonta Fer.) 1. Testa cylindrica. 62. Helix C. anconostoma, Prodr. MS'—Tab. 6. f. 30.
H. testa cylindrica, pupzformi, leviuscula, nitida, corneo-rufes- cente: spira obtusa; anfractibus convexis, rotundatis, quis, striis transversis, obliquis, obsoletis, indistinctis; sutura distincta, impressa: apertura 1-dentata, elliptica, sub-angustata, longiore quam lata, sub-tri- gona, anticé angulata: columella recta, supra cubito vel flexura abrupto, acuto, cum labro tenui, reflexo conjuncta: dente lamellato in ventrem juxta labrum obsoletiusculo, 4 labro distincto.
a, gyrata; testa elongata: aperture cubito distinctissimo. f. 30. 1. ¢.
Long. 13 lin. Diam.1. <Anfr. 7.
B. curta; testa abbreviata: aperture cubito obsoletiore.
Long. 14 lin. Diam. 3. Anfr. 6.
Hab. in Madera.
Helici C. umbilicate, Fer. (Pupa umbilicata, Drap., Lam*.; Turbo muscorum, Mont. t. xxii. f. 3.) proxima, presertim per varietatem (ra- riorem) 8. curtam; sed distincta videtur. In Icone cl: Montagui su-
s
pra indicata, testa Britannica ejusque characteres 4 nostra Maderensi optimé distinguuntur.
of Madera and Porto Santo. 63
63. Helix C. cheilogona, Prodr. MS.—Tab. 6. f. 31.
Hi. testa sub-ovata, cornea, levi, vel obsoleté striata: apertura 3-plicata, coarctata, anticé prominula; plica unica in columellam; duabus parallelis in ventrem positis; intermedio minore: labro ex- panso, ints marginato, sinuato-angulato: umbilico magno, patulo, pro- fundo.
Long. 12 ln. Diam. 1. Anfr. 6.
Hab. in Madera.
64. Helix C. sphinctostoma, Prodr. MS.—Tab. 6. f. 32.
HT. testa cylindrica, fusca: anfractibus planis transversé sub-stri- atis: apertura 4—6-plicata; plicis duabus in columellam, postica obso- leta; duabus in ventrem, quarum anterior plice anteriori columellari zqualis, posterior magna, complicata, cum dente ad angulum inferiorem labri posito in unum conjuncta: labro reflexo, posticé sub-angulato vel sinuato, ad angulum intts dentato, anticé 1-2-plicata : umbilico pa- tulo, profundo.
Long. 2 lin. Diam. 1 Anfr. 7.
Hab. in Madera.
Testa plus minis striata.
65. Helix C. monticola, Predr. MS.—Tab. 6. f. 33.
HI. testa cylindrica, castanea, pallido fasciata: anfractibus con- vexis, tumidis, striis elevatis, quidistantibus, transversis sculptis ; sutura impressa: apertura sub-sexdentata; columella 92-plicata, _plica posteriore obsoletissima; plicis duabus approximatis, parallelis, in ven- trem, gquarum anterior minor; posterior magna, cum labro continua: labro sub-reflexo, wquali, 3—plicata; plica intermedia majore; anteriore et posteriore minutis.
Long. 13 lin. Diam. vix 1. Anfr. 6.
Hab. in summo cacumine Montis “Pico de Facho” Insule Por- tus SY.
Oss. Priori (H. sphinctostomati) quoad plicas affinis; sed distincta.
64 Mr. Lowe on the New Plants and Land Mollusca
66. Helix C. calathiscus, Prodr. MS—Tab. 6. f. 34.
HZ. testa cylindrica, ovoidea, abbreviata, castanea, pallido fasciata: anfractibus convexis, sub-tumidis, costulis aquidistantibus, transversis, crebris, sculptis; sutura impressa: apertura sub-septemplicata; colu- mella 1-plicata; plicis duabus in ventrem, quarum anterior valde in- terna, minuta, dentiformis; altera posterior magna, cum labro continua: labro expanso, sub-sinuato; callo ints margini parallelo, postic® in dentem duplicem desinente, anticé dente minuto, simplici, obsolete et plica unica intermedia, magna, instructo.
Long. 17 lin. Diam. 1. Anfr. 7.
Hab. in summo cacumine montis “ Pico de Facho” Portis S*.
67. Helix C. cassida, Prodr. MS\—Tab. 6. f. 35.
H. testa ovata, ventricosa, abbreviata, sub-imperforata: anfractibus planis, striis elevatis, crebris, aquidistantibus, transversis; sutura sub- indistincta: apertura 7—8-plicata; columella biplicata, plica posteriore minore; plicis duabus sub-zqualibus, parallelis in ventrem positis, ex- teriore paulld majore, cum labro continua, sinum efficiente: labro ex- panso, 5-plicata; plica anteriore minore, aliquando obsoleta; tribus in- termediis lateralibus, approximatis, superiore magna, duabus inferioribus minoribus, quarum infima sub-dentiformis; quinta infima minima, ad angulum labri posita: perforatione minima.
Long. 2 lin. Diam. 13. Anfr. 7—8.
Hab. in Maderz convallibus, in rupibus aridis umbrosis.
Recens semel tantum lecta; necdum vivam vidi. Ad locum “Canical” dictum, inter alias plurimas Helicis species* paulld frequen- tior, sed statu semifossili. ;
* Teste illa, hie et in Portu S*., statu semi-fossili, in arena calcareA, inter concreta ramiformia (minimé “ Lignites”), reperte, omnes terrestres, plurime (forsan omnes) etiam hodié in Madera vel Portu S®. yivunt; nec ullam quidem speciem marinam cum illis com- mixtam vidi. “ Delphinula sulcata Lam*.?” Bond. Exc. p. 140. f. 33. a, b, est Helicis species, (Helix Delphinula nob.) valde elegans, Helici tectiformi Sow. affinis. A Delphi- nul& prorsts aliena.
of Madera and Porto Santo. 65
VI. Genus, CLAUSILIA, Drap’. Helicis sub-genus Cochlodina, Fer. 68. Clausilia crispa, Prodr. MS.—Tab. 6. f. 36.
C. testa turrita, sub-ventricosa: anfractibus convexiusculis, striis transversis, creberrimis, minutissimé flexuosis sculptis, interstitiis elegan- tissimé decussatim punctulato-striatis, quasi cancellatis; sutura distincta, impressa: apertura oblonga, biplicata; plicis columellaribus, approxi- matis, divaricatis, sub-posticis, postica prominente sc. extrorsiim ad mar- ginem peristomatis producta eique continua, sinum ad angulum posti- cum aperture formante: peristomate simplici, acuto, sub-expanso: costis dorsalibus rimaque umbilicali obsoletis.
Long. 7 lin. Diam. 2. Anfr. 9.
Hab. in rupibus sylvarum Madere.
Peristomate nec elevato neque disjuncto sc. columellari obsoleto, necnon costis duabus dorsalibus rimaque umbilicali sub-nullis, quin et quodammodo forma et magnitudine ad Clausiliam bidentem Drap. (Turbinem laminatum Mont.) magis quam ad aliam quampiam speciem accedit.
69. Clausilia deltostoma, Prodr. MS.—Tab. 6. f. 37, 38.
C. testa turrita, gracili, obtusa: anfractibus planiusculis. striis rec- tiusculis, crebris, elevatis sculptis: apertura oblique obovato-rotundata, deltoidea, effusa, posticé angustata, sub-biplicata; plica antica columel- lari, interna, obliqua, duplici; postica simplici, prominente sc. extror- sum ad marginem peristomatis producta eique continua, sinum ad an- gulum posticum aperture efficiente: peristomate continuo, expanso, reflexo, disjuncto.
Long. 5—5} lin. Diam. 14. Anfr. 10—11.
a. anfractibus convexiusculis; sutura distincta.—f. 37. |. c.
Hab. in Madera et Portu 8S”.
B. anfractibus planatis; sub-obsoleta.—f. 38. 1. c. Hab. in Madera. Vol. IV. Part I. I
66 Mr. Lowe on the New Plants and Land Mollusca, §c.
Clausilie labiate Sow. (C. bicanaliculate sec. Fer.) nimis affinis; sed tripld minor; gracilior; anfractibus, etiam in £., minus planis; sutura mints obsoleta; plica posteriore apertures prominente, margini labri continua, nec interna; peristomate minis incrassato, nec labroso. Variantur etiam et a. et 6. collo aperture magis minusve producto. Clausilia retusa (Bulimus retusus, Oliv.) etiam forma et habitu magis affinis; sed striolis exilissmis aliarum interstitia decussantibus differt. 70. Clausilia exigua, Prodr. MS.—Tab. 6. f. 39.
C. testa parvula, turrita, gracili, obtusa: anfractibus planiusculis, omnibus transversé creberrimeé striatis: apertura obliqué obovata, _bi- plicata; plica antica valde interna; postica prominente, margini peris- tomatis producta, continua, sinumque cum labro efficiente; ambabus
columellaribus, simplicibus: peristomate continuo, reflexo, posticé sub- sinuato.
Long. 3—3} lin. Diam. 1. Anfr. 8. Hab. in Madera.
Clausilie parvule Leach, ut videtur, affinis.
3. Familia, Cyclostomide. VII. Genus, CYCLOSTOMA, Lam‘. 71. Cyclostoma lucidum, Prodr. MS.—Tab. 6. f. 40.
C. testa globoso-conoidea, nitida, leviuscula, sub-imperforata: an- fractibus convexis, transversé sub-striatis; sutura impressa. Axis 2 lin. Diam. 2}. Anfr. 5.
Hab. in Madere humidis sylvaticis.
Testa, quoad formam, Valvatam piscinalem referens, fusca, olivaceo- cornea, lucida. :
is
ng
ff. f. f. ff. f
£12:
2 3 4. 5
OS Di
TABULARUM EXPLICATIO.
Tas. I.
Prant of Goodyera macrophylla, nob. natural size.
A single flower with its germen and bractea.
The uppermost of the three outer petals of the perianth. Two lowermost of ditto.
The uppermost of the three outer and the two inner petals of the Peri- anth; the three cohering upwards by means of the former at the back of the two inner.
The same two inner petals, separated.
Side view of f. 5.
Labellum with column and anther-case, in situ. Side view of the same.
Same as f. 9, with the anther-case lifted up (artificially) shewing the two Pollen-masses in situ.
Inside of anther-case, shewing the dissepiment. Front view of the two Pollen-masses, in situ.
All the figures, except f. 1, more or less magnified.
Tas. IT.
Plant of Tolpis crinita, nob.; smoother and with more entire leaves than usual. The more common state of the same. A seed with its pappus; magnified. 12
68
f.
2.
12.
a i ct Sie TS ll
29 S
i _
Tabularum Explicatio.
Tas. III.
Branch of Sedum fusiforme, nob.
A single petal with its gland, and stamen; magnified.
Tas. IV.
Branch of Ononis dentata, Sol.
Vitrina Lamarckii, nob. \b, state of the same in which the yolutions are
Helix
Tas. V.
a, common state.
visible internally to the apex. furva, nob, var. a. erubescens, mob. var. a. sub-plicata, Sow. [Testa (“ viva” dicta) junior| undata, nob. ; phlebophora, nob. arcta, nob.—magnified. fausta, nob.—ditto. arridens, ob.—ditto. Webbiana, »ob.—Two views of same individual. Bulveriana, Wood,—ditto, ditto, tectiformis, Sow. subtilis, 0b,—magnified. actinophora, mob. Porto-Sanctana, Sow. var. a, nob. »—. — B. nob, pusilla, 20b.——magnified, bifrons, nob. paupercula, ob.—magnified.
a, state in which it is found, coated with soil.
t: le obtecta, nob: is, the same cleaned.
dealbata, nob. var. a. granulata.
Pho kh rh te bh bh Eh Po bh th rh fe fh rb bh Thoth bh rh ph ph be be be fs
_
Se GP Ss ie Tos aS aes
Helix
Tabularum Explicatio.
maderensis, Wood. compar, ob.—magnified. leptosticta, 20b.—ditto. lentiginosa, »ob.—ditto.
calva, nob,
Tas. VI,
abjecta, nob.
compacta, nob.
consors, 70b.
depauperata, nob.
lurida, nob.
nitidiuscula, Sow.
punctulata, Sow., var. a. nob, » ——» — £B.-nob.
lauta, mob.
rotula, mob.
polymorpha, mob. var. a. irrasa. » — —. B. depressiuscula. > — —. ¥y- arenicola. eee SO » — —. e. calcigena.
> — — ¢. pulvinata, cheiranthicola, nob. Subvar. 1. zonata, oxytropis, nob. echinulata, ob,—magnified, duplicata, ”ob.—ditto, turricula, nob, bicolor, nob.—magnified, C. tornatellina, ob.—ditto, C, melampoides, mob.
C. triticea, nob. var. a. biplicata,—magnified,
69
J S
2 >
2.
Tabularum Explicatio.
Helix C. triticea, mob. var. 3. edentula—magnified. . ovuliformis, ob.—ditto.
. gracilis, nob.—ditto.
. lubrica, Mull. var. nob.—ditto.
. anconostoma, nob. var. a.—ditto.
. cheilogona, nob.—ditto.
. sphinctostoma, ob.—ditto. . monticola, 2ob.—ditto.
. calathiscus, 2ob.—ditto.
. cassida, nob.—ditto.
ol @) @) ©) ©) Tele eke mie!
Clausilia crispa, nob.—ditto. deltostoma, nob. var. a.—ditto. —__———, 700. —. (3.—ditto exigua, nob.—ditto.
Cyclostoma lucidum, ob.—ditto.
Il. On the General Equation of Curves of the Second Degree.
By AUGUSTUS DE MORGAN,
OF TRINITY COLLEGE, CAMBRIDGE,
AND PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF LONDON.
[Read November 15, 1830.]
Tue object of this Paper is to draw attention to some properties of Curves of the Second Degree, by means of which the reduction of their equations from one set of axes to another is materially facilitated. Little, if any, notice of these properties has been taken, nor do I remember to have seen their existence mentioned with the exception of two very limited particular cases, viz. that the sum of the squares of conjugate diameters, and the parallelogram formed by them are constant.
Suppose that any curve of the Boeone degree is referred to axes which make an angle @ or that vy = 0. Suppose the ori- gin removed to a point whose co-ordinates are m and n in the directions of x and y, and moreover suppose the directions of the axes to be changed so that t= 9% sip nae x and y’ being the new co-ordinates; and let hy =~-—o=6. Let the equations of the curve referred to the first and second systems of axes be
ay +bxy tex +dy +ex +f =0, (1)
a ay? +bvy +e a "yi +éut+f'=0; (2)
72 Mr. De Morean on the General Equation of Curves
From the suppositions made respecting the axes wih sin (0—@) x sin (@—W)
pes i sin @ 3 sin 0 YT _ sing, sin fa (3) = tie, Ri
If we substitute in (1) these values of x and y, the resulting coefticients of y”, xy’, &c. must be proportional to a’, b’, &c. Pre- serve this condition and make the substitutions and developments
as follows: Let A =a—hcos 0+€¢ cos*6, B =sin 0 (b— 2c cos 9), C =csin’ 8,
4 D=2an+bm4+d—cos 0(2cm+bn+e), (4)
E =sin 0(2cm+bn+e), F =an'+bmn+cem+dn+em+/f.
In which it is important to observe that if the primitive co-ordinates be rectangular 4=a, B=b and C=c. If in addition the origin be not changed D=d, E=e, F=f. Also that if the axes of x’ is parallel to that of « and wy’ = 90°, the equation becomes
Ay + Bry + Ca’ + Dy + Ex+F=0. The substitution above indicated will now give the following
results :
ra! sin? @=A sin? + B sin cos y + € cos’ y,
rH’ sin?@=2 A sin y sin p+ B (sin yy cos p t+cos v sin p) +2 C cos ¥ cos p,
Ac’ sin? @= A sin? p+ Bsin ¢ cos p + C cos , (5)
rd’ sind =D sin +E cos yy,
re sin 9 =Dsin p+ E cos ¢,
Af’ =) where \ is any quantity whatever.
of the Second Degree. 73
From which we find A*(b°—4a'c’) sin’ @=(B* — 4 AC) sin® (—) = (b* —4ac) sin® @ sin? 6 b°—4a'c —s B—Aace 2 =— — Mee sin? @ (6) of which one particular case is that the parallelograms described about conjugate diameters are always the same.
Again we find that ad+c'—b' cos _ a+ce—b cosé
7 (7)
sin’ @’ sin? @
of which when divided by (6) one particular case is that the sum of the squares of conjugate diameters is always the same.
When
a+e—bcos@=0.
it indicates an equilateral hyperbola. Again, we find that 3cd*t+a'e"—b'd'e' _ cd’+ae*—bde—(b*—4ac) (an°+bmn+cem'+dn+em) ) sin® 6 a sin® @ ( from (8), (6) and the last of (5) we deduce that ed°+ae?-Ude cd’+ae’—bde x 1) = tei ay
b?—Aa'e’
(9)
If each side of this be nothing, the equation represents, either two straight lines which intersect, a point, or is impossible.
If the new co-ordinates be such that 6’=0 we find from the second of (5) 24 tan ¢ tan y+ B(tan f + tan W)+2C=0. If in addition to this the new axes must be rectangular, in which case tan ¢ tan ¥+1=0, we find the following equation
sin 0 $b—2c cos 0}
— : 10 a—b cos 6 +c (cos’ 0—sin* 6) (ao)
B
from which two values of y are found differing by 90°, either of which may be the value of ¢, and the other of ¥. Vol. IV. Part I. K
74 Mr. De Moraan on the General Equation of Curves
The lines determined by these angles are parallel to the principal diameters. The next question is, how can the greater principal diameter be distinguished from the lesser? It must be observed that in the general equation
a _ rectangle of segments of the axis of x _ (diameter parallel to axis of «)’ c rectangle of segments of the axis of y (diameter parallel to axis of y)?”
Since the square of one or both of these diameters may be negative, that which is numerically the greater can be ascer- tained only when we know the sign of c*—a’. If this be positive the greatest of the two diameters above-mentioned is parallel to the axis of y, &e.
In (10) one of the values of x must be less than 90°. Take this for the new axis of x, that is, let sin 2¢ be positive. From (10) it appears that
C-A
5 B sin 2¢= 7 cos 26 = yy (11) where M=+.,/B’+(A-—C)*, and since sin 2¢ is supposed positive, M must be of the same sign as B. Also sin )=cos ¢, cos y= -sin #, since y—@=90°. Therefore from (5) ra’ sin’ =A cos* d—B sin } cos P+ C sin’ 4, re’ sin’ @=A sin’ +B sin p cos P+ C cos’ g, A(c' +a’) sin? A= A+ C, d(e’—a’) sin? @=(C— A) cos 264+ B sin 26=M from (11).
Therefore c’—a’ has the same sign as M(4 +C) BIAS),
since B and M have the same signs; the hypothesis bemg that the principal diameter which is nearest to the axis of x in the positive direction is parallel to the axis of x’. Accordingly there- fore as B(4+C) or sin 6 (b—2c cos@)(a+e—bcos#) is positive or
of the Second Degree. 75
negative, the less or greater principal diameter is nearer to the axis of x.
When B(4+C)=0 there are two cases to be distinguished. If 4+C=0 the curve is equilateral, but can only be an hy- perbola. If B=o0 and C—A is not =o from (11), one of the principal diameters is parallel to the axis of x. Which it is may be determined from the sign of c*— a’.
When B=o and C—A=0, or which is the same, b=2c cos and a=c, the position of the principal diameters is indefinite : that is, the curve is a circle.
To determine the magnitude of the principal diameters, the equation must be reduced to the form
ray’ + Ac'x* +f’ =0, where 6 = 90°.
In this case the equations (6), (7) and (9) become
—ANa'c' = Le! 2 sin°@ , , —b isa _ate cos @
sin’ @
cd’ + ae’— bde Ni b?—Aac ay
Whence the squares of the principal semidiameters or = ;
and — 7 are contained in the formula
cd’ +ae’*—bde b?—Aac i
— 2 sin? Qn a+c—bcos6+,/(a—c)*+b°—2 cos 0(a+c)b—2ac cos)
In this way the curve might be referred to the conjugate diameters which make a given angle 6’, and the limits of the value of & might be determined.
K 2
76 Mr. De Moraan on the General Equation of Curves
When the asymptotes are the axes of co-ordinates, the equa- tion of the hyperbola is \0'2'y7/ + Af’= 0.
The position of the asymptotes is then determined from the equation A tan? x + B tany + C=0, sind b—2ccos@+./b?—4ac 2 ° a—bcosé+ccos*d ”
or tany =
where the two values of x are those of ¢ and y¥.
The equations (6), (7) and (9) become Nb? _ b —4ae sin?6’~—s sin? @
rU'cos8’ — a+c—hcosd
sin? 0 sin? 0
,_ ed’ +ae—bde ae Ne aimedarye
G i bP whence tan@’= + Bin ake 4 ae
a+c—bcosé and the equation is b’—4ac ,,, ed? +ae?—bde = J(a=0)' +b —2c0s0(ate)b—2aecos0) % * b’—Aac vas In referring to the expression for the principal semidiameters, it
cd’+ae>—bde ~ b=4ae +7? the greater diameter is possible, or the curve lies in: the acute angles of the asymptotes, and the contrary.
appears that if a+c—b cos @ has the same sign as
If the equation of the parabola be reduced to the form a'y*\+éa' =0, where 0=90°, the equations (7) and (8) take the following form
ee a+ce—bcos@ a—2./accosd+e i, sin? @ Th sin? @
of the Second Degree. 77
ed'+ae’—bde _ (/cd—,/ae)’
sin? @ sin’ @
Nae? =
whence the equation becomes
en Jae
sin? @x=0, ¥tqr 2,/accos0 +c)! 3
The position of the axis is determined by the same criterion as that for finding the major axis of the ellipse; and the curve may be further ascertained if we recollect that it must lie en- tirely on one side of the right line whose equation is dy + ex+f=0, and on that side in which the co-ordinates of a point are such as to make dy+ex+/ of a different sign from a or ec.
In order to find the co-ordinates of the vertex, we must have recourse to equations (5) recollecting that B’-—44AC=o0, and that tang@tany+1=0. From the third and fourth of these we find
ies Aa Ja—,/ecos 0
Tire kil 6: Geno, But the fourth and fifth of (4) give the following equation D_ 2,/ak+d—cos0(2,/ck+e)
EE (2,./cek+e)sin@ where k=,/an+./em
and the last of (4) gives k?+dn+em+f=0,
from the two first of which we find
= (e,/a+d/c) cos 0—(d,/ate,/c) k= + {+N ee : Jan sfem= a—2,/accosd+e
dn+em=—f—k’,
and from the last
from which equations m and x may be found. The expressions present nothing remarkable.
73 Mr. De Morean on the General Equation of Curves, &c.
The consequences of these formule might be carried further, but what has been here said may be sufficient to turn the attention of elementary writers to this part of the subject. Analogous formule may be obtained for Surfaces of the Second Degree, which will probably form the subject of a future communication.
AUGUSTUS DE MORGAN.
III. On the Nature of the Light in the Two Rays produced by the Double Refraction of Quarts.
By G.B. AIRY, M.A.; M.G.S.;
LATE FELLOW OF TRINITY COLLEGE; PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY IN THE UNIVERSITY OF CAMBRIDGE:
AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.
[Read February 21, 1831.]
I propose in this paper to offer some conjectures as to the nature of the light forming the two rays produced by the double refraction of quartz; to describe the experiments on which they are founded ; and to explain the calculations by which the theory and the experiments are compared. The subject is one to which (I believe) no attention has been paid, except by one distin- guished foreigner; the mode of calculation is original to me. and is, to the best of my knowledge, new.
It is well known that the rays produced by the double re- fraction of calc spar, (calcareous spar, Iceland spar, or rhom- bohedral carbonate of lime) and most other doubly refracting crystals, are entirely polarized: one in the principal plane pass- ing through the ray (or, if a biaxal crystal, in the plane equally inclined to the planes passing through the ray and the two axes) and the other in a plane perpendicular to the former. From the exact agreement of the phenomena of depolarization with the calculations made on this hypothesis, we are justified in supposing that the law holds true when the rays are so little
80 Proressor Airy on the
separated that it is difficult to observe them in the common mode of inspection. Now it has generally.been supposed that the two rays of quartz are polarized in the same way: differing from those of cale spar only in the magnitude and direction of their separation. It was known however almost as soon as Arago and Biot commenced their observations, that there is some anomaly in the rays passing in the direction of the axis of quartz; and the latter of these observers established the difference of right-handed and left-handed quartz. Fresnel by a simple experiment* (which I have repeated) shewed that the light in the direction of the axis of quartz is not one ray, but two rays moving in the same direction, and with different ve- locities. He shewed moreover that a new kind of light may be produced by causing polarized light to undergo two internal reflections in a glass rhomb with certain angles, the plane of polarization bemg inclined to the plane of incidence at an angle of 45°; and that this light is exactly similar to one or other of the two rays abovementioned according as the plane of polarization is on one or the other side of the plane of in- cidence. And by a mathematical investigation, of which I am unable to supply the deficient steps, he shewed that the effect of the internal reflections is to retard by one quarter of an un- dulation the undulations perpendicular to the plane of incidence, so that in the light thus modified the particles of ether which were originally in a straight line will at any time be found in the form of a circular helix, and each will revolve ‘uniformly in a circlet. And from the nature of the original experiment
* It is not easy to make this experiment in a satisfactory manner. If the axes of the crystals are not precisely adjusted, several images will be seen. I have not succeeded in obtaining two only, though I haye made the others much more faint than the two principal.
+ As I cannot appreciate the mathematical evidence for the nature of circular polarization, I shall mention the experimental evidence on which I receive it. 1%. The light when re-
ceived
Double Refraction of Quartz. 81
it appeared that in right-handed quartz it is necessary to sup- pose the right-circular polarization transmitted with the greater velocity; in left-handed quartz the contrary. I have repeated and varied most of Fresnel’s experiments relating to this sub- ject, and am perfectly convinced of the correctness of his views.
Now if, in the experiment with the glass rhomb, the planes of polarization and incidence be inclined at any other angle than 45°, the magnitudes of the undulations parallel and _per- pendicular to the plane of incidence will no longer be equal: but the alteration of their periods will be the same as before. The displacement of the particles of ether will still be repre- sented by a helix, but instead of being traced round a circular cylinder, it must be supposed traced round an elliptic cylinder. This modification may properly be called (as Fresnel has called it) elliptical polarization*. This term has since been used by Dr. Brewster to express the nature of the light (probably iden- tical with this, or nearly so) reflected from metallic surfaces.
ceived on an analyzing plate or tourmaline presents the same appearance in whatever direction the analyzing plate is turned round the incident ray. 2%. The phenomena of depolarization are the same in whatever direction the analyzing plate is turned. 3". If the polarized light passes through two such rhombs placed in similar positions, the plane of polarization is shifted 90°. 4. If they are placed in crossed portions the plane of polarization is unaltered. 5. The phenomena of depolarization agree with the calculations founded on this supposition : in uniaxal crystals, where the plane of polarization of one ray is changed 360° in going round the axis, the alternate quadrants are pushed in and thrust out one quarter of a tint: and in biaxal crystals, where the plane of polarization is changed only 180° in going round the axis, the alternate semicircles are altered in the same manner.
* If I might venture to fix on the discovery of Fresnel, which among all his wonderful ad- ditions to optical science appears likely to possess the greatest practical value, I should select his invention of the mode of producing circularly-polarized or elliptically-polarized light by internal reflexion of plane-polarized light in glass or water. He has given us the power of producing light whose laws ure as well known as those of plane-polarized light, and which is more manageable, inasmuch as it admits of degrees in its ellipticity. The beautiful geometry of Malus is forgotten when we think of the discovery of polarization: the far more valuable theoretical discoveries of Fresnel will lose their preeminence when put in competition with an invention which enables others to make discoveries.
Vol. IV. Part I. 1,
82 Proressor Airy on the
Should any difference be found, I have no hesitation in fixing on the modification above described as that to which it ought in propriety to be attached.
I am now able to explain my conjectures on the nature of the light in the two rays of quartz.
1. I suppose the ordinary ray to consist of light elliptically polarized, the greater axis of the ellipse bemg perpendicular to the principal plane; and the extraordinary ray to consist of light elliptically polarized, the greater axis of the ellipse being in the principal plane.
2. I suppose that when the ordinary ray is right-elliptically- polarized, the extraordinary ray is left-elliptically-polarized : and vice versa.
3. I suppose that the proportions of the axes of the two ellipses are the same: each proportion being one of equality when the direction of the ray comcides with the axis, and be- coming more unequal, according to some unknown law, as the direction is more inclined to the axis: the minor axes of the ellipses having sensible magnitudes when the rays are inclined 10° to the axis. '
4. I suppose that the course of the rays after refraction can be determined by the construction given by Huyghens for calc spar, with this difference only, that the prolate spheroid for de- termining the course of the extraordinary ray must not be sup- posed to touch the sphere for determining the course of the ordinary ray, but must be entirely contained within it.
These conjectures were originally suggested by the desire of finding some connecting link between the peculiar double re- fraction in the axis discovered by Fresnel*, and the double
* It does not appear, I think, that Fresnel had made any distinct supposition as to whether the two rays in the axis should be considered as the ordinary and extraordinary ray in their
ultimate
Double Refraction of Quartz. 83
refraction commonly recognized. All the phenomena of colours which I have observed agree perfectly with the results of my hypotheses.
I may mention that I have found observations corresponding to many detached parts of the phenomena which I have viewed assembled, in the early memoirs of Arago and Biot. But the method which these philosophers used, (particularly the latter), of examining a small part only at a time, does not appear to be well adapted to the discovery of the laws of light. In the experiments which I am about to describe, every thing depends on the form of the coloured curves; and to attempt to dis- cover this from observation of detached parts would be perfectly hopeless. These coincidences I have recognized only since I made my own observations.
It must be observed that all the phenomena mentioned be- low are described as they appear when examined with an ana- lyzmg plate of unsilvered glass. If a plate of tourmaline be used, the right and left parts of the image will have the same relative position, but the upper and lower will be interchanged : the observer’s eye being supposed to turn in such a manner that the axis of the tourmaline appears wp and down.
PHENOMENA.
I. If a plate of cale spar cut perpendicular to the axis be examined with the polarizing and analyzing plates crossed, the system of rings is that represented in fig. 1. If the analyzing plate be turned less than 90° either way round the incident ray, the system of rings is that represented in fig.2: and if turned *
ultimate state, or not. From all that I could extract from his Memoirs, I was always in doubt whether both the ordinary and the extraordinary ray in the neighbourhood of the axis ought not to be considered as divided each into two circularly polarized rays.
LQ
84 Proressor Airy on the
exactly 90°, it is that of fig.3. The order of colours does not sensibly differ from Newton’s scale, beginning with black. These are common and well known phenomena.
II. If Fresnel’s rhomb of glass, mounted as represented in fig. 4, be placed to receive the polarized light, so that the plane of reflection pass through the divisions 45° and 225°, the calc spar will present the appearance of fig. 5. The rings are abruptly and absolutely dislocated: those in the upper right- hand quadrant and the quadrant opposite to it are pushed from the center by one-fourth of an interval, and those in the other quadrants are drawn nearer to the center by the same quantity. The line separating the quadrants is no-where black: the in- tensity of its light is uniform and about equal to the mean intensity. If the plane of incidence pass through 135° and 315°, the phenomena of adjacent quadrants are exactly interchanged. No alteration is made by turning the analyzing plate round the incident ray: the lines dividing the quadrants are always pa-. rallel and perpendicular to the plane of reflexion at the ana- lyzing plate*.
Ill. If the plane of reflexion in the rhomb pass through 0° and 180°, or through 90° and 360°, the phenomena are precisely the same, and undergo the same changes as those in Pheno- menon I. If while the plates are crossed the rhomb be turned gradually from the position 0° towards 45°, the rings are gradu- ally changed, at first becoming (as far as the eye can judge) elliptical, and then assuming the form represented in fig. 6.
IV. If a plate of quartz, whether right or left-handed, be
* It is proper to mention that I had exhibited this phenomenon to the Cambridge Philo- sophical Society in the spring of last year, long before the publication of Dr. Brewster's valuable Memoir in the Phil. Trans. for 1830, and (I believe, but I do not recollect the date) before its communication to the Royal Society.
Double Refraction of Quartz. 85
interposed between the crossed plates, a set of rings is seen as in figs 7, 8, 9, 10. As far as the eye can judge, the rings are ex- actly circular, but there is no black cross, and the central tint is not black, but removed from it by a number of tints in Newton’s scale proportional to the thickness of the quartz. Thus with a thickness 0,48 inch, the central tint is pale pink: with a thickness 0,38 inch, the central tint is bright yellowish green : with thickness 0,26 inch, it is a rich red plum-colour: with thickness 0,17 inch: it is a rich yellow.
The colours then appear to be nearly the same, beginning from the center, as in Newton’s scale, beginning with the tint representing this central tint. At a considerable distance from the center four dark brushes begin to be visible, in the same directions as the arms of the black cross in calc spar.
V. Now (supposing the crystal right-handed), if the plate of quartz be thin, and the analyzing plate be turned, the upper part towards the observer’s left hand, a blueish short-armed cross appears in the center*, which on turning further becomes yellow: and the rings are enlarged. On turning still further, the cross breaks into four dots. The rings are no longer cir- cular, but of a form intermediate between a circle and a square, their diagonals (as well as the cross) being inclined to the left of the parallel and perpendicular to the plane of reflexion. See fig. 11. If the analyzing plate be turned the other way, there is no cross: the form of the rings is changed. from circular nearly as in the former case.
VI. If the plate of quartz be thick, the dilatation of the rigs and the change of form are all the perceptible phenomena.
* This may be considered as the definition of right-handedness of the crystal: and this observation gives the readiest means, with a thin plate, of determining whether it is right- handed or left-handed. If the plate be thick, the easiest method is to observe in which direction the analyzing plate must be turned to make the rings dilate.
86 Proressor AIRY on the
And on turning the analyzing plate continually to the left, the rings continually dilate, and new spots start up continually in the centre, and become rings. If the crystal be left-handed, the remarks in this and the last article apply equally well, sup- posing the analyzing plate turned in the opposite direction.
VII. If Fresnel’s rhomb be placed in the position 45°, and the light thus circularly polarized pass through the quartz; on applying the analyzing plate, mstead of rmgs there are seen two spirals mutually inwrapping each other as m fig. 12. If the rhomb be placed in position 135°, the figure is turned through a quadrant. If the quartz be left-handed, the spirals are turned in the opposite direction. The central tint appears to be white. With the rhomb which I have commonly used (which is of plate glass, but with the angles given by Fresnel for crown elass) there is at the center an extremely dilute tint of pink: I think it likely that this arises from the error in the angles, as the intensity of the colour bears no proportion to that m other parts of the spirals. The figure was drawn from the ap- pearances given by a plate of quartz 0,26 inch thick.
VIII. If two plates of quartz of equal thickness, but cut one from a right-handed and the other from a_ left-handed crystal, be attached together, and put between the polarizing and analyzing plates, the left-handed slice nearest to the polar- izing plate, the appearance presented is that of fig. 13. Four spirals (proceeding from a black cross in the center; which is inclined to the plane of reflexion) cut a series of circles at every quadrant. The points of intersection are in the plane of re- flexion, and perpendicular to it. This is the simplest way of describing the form: but if we followed the colours which gra- duate most gently, we should say that the form of each is alternately a spiral and circular are, quadrant after quadrant.
Double Refraction of Quartz. 87
Ata distance trom the center the black brushes are seen. If the combination be turned so that the right-handed slice is nearest to the polarizing plate, the spirals are turned in the oppesite direc- tion. This is one of the most beautiful phenomena of optics. The slices from whose appearance the figure was drawn are each 0,16 inch thick.
I shall now proceed to explain the mode of calculating these phenomena on assumed Jaws of the nature of light in the two rays of crystals.
In fig. 14, let 468, CD, be two parallel rays of the same pencil incident on a plate of cale spar cut perpendicular to its axis, of which one furnishes the ordinary ray BH, and the other the extraordinary ray DE, which afterwards pass in the same direction EF. (The extraordinary ray of AB, and the ordinary ray of CD are not to be considered here, as they do not emerge at H: but each of them will interfere with some other ray). The paths are found by this construction. Draw GK a tangent to a circle whose radius is GHx (preserving Biot’s notation); draw LN a tangent to the ellipse whose semi- axes are LM xa, LMxb. The velocity and direction of the ray will be represented by the radius joining the point of in- cidence with the point of contact, the velocity m air being represented by GH, LM. Hence the path of the ordinary ray (measured by the path in air which it would have described in the same time) exceeds that of the extraordinary by
GH LM
Putting 9 for the angle of incidence, and 7' for the thickness of the plate, this is found (after all reductions) = = 1./1—6 sin? 0—./1—a' sin’ 6}.
When 6 is small, this is nearly = 7’ x ee x 6. Call this 0.
88 Proressor Airy on the
I shall suppose with Fresnel, that by a ray polarized in one plane is meant a ray whose vibrations are entirely perpendi- cular to that plane; that consequently the vibrations forming the ordinary ray in the crystal are entirely perpendicular to the principal plane passing through that ray, and that those form- ing the extraordinary ray are wholly parallel to that plane.
I*. Now suppose a pencil of polarized light to fall with a small angle of incidence on a plate of cale spar cut perpendi- cular to its axis. Let us conceive ourselves looking in the di- rection of the incident ray; and let fig. 15. represent the pro- jections (on a plane perpendicular to the incident ray) of the planes of polarization of the polarizing and analyzing plate, and of the principal plane of the crystal passing through the inci- dent ray. Let P:Ap, the plane of polarization (or of reflection) at the analyzing plate, make an angle « with P,Ap, the original plane of polarization; and let CAc the principal plane of the crystal passing through the ray make the angle ¢ with the former plane. The displacement of the particles of ether pro- duced by the wave as originally polarized, may be represented
by c.sin an (vt—x), where is the interval of space between two
waves, « the distance measured from any arbitrary point, ¢ the time since the ether at that point was at rest, and v the velocity of the wave. (This applies even after the wave has passed through any media, provided we take for « the space which would in the same time have been described in air). And this dis- placement is entirely perpendicular to P,p,. This may be re-
solved into e.sin = .(vt — x).cos (a + ¢) perpendicular to AC, and
* The Phenomena, and the investigations corresponding to them, are numbered in the same way.
Double Refraction of Quartz. 89
¢.sin = (vt—.x).sin (a+) parallel to 4C. The former of these
furnishes the ordinary ray, the latter the extraordinary. Now after they have passed through the crystal, we may still keep the expression for the vibration of the ordinary ray, provided we make the proper alteration in the value of «: but (by what has gone before) we must then suppose the path of the extraordinary ray shorter by ©. Consequently after passing through the cry- stal, the vibration produced by the ordinary ray is
c.sin <7 (vt—2) -cosa+q@ perpendicular to AC, and that produced by the extraordinary ray is c.sin 27 (vt—a+ 8).sina+@ parallel to AC.
When these are received on the analyzing plate, those parts only are transmitted to the eye or to the screen, which are perpen- dicular to AP;. They are
a == e.sin * (vt 2). cosa +p. cos
and c. sin <™ (vt—# + 0). sin a+@.sin @:
and the sum of these represents the magnitude of the vibration which comes to the eye, or falls on the screen. It may be put under this form
sin —* (v¢—2) {e.cos a+ p.cos p+¢.cos " O.sin a+ @.sin 9}
+ cos *2 (vt—«).c.sin =7 ©. sin a+@.sin d.
Vol. IV. Part I. M
90 Proressor Airy on the
Now it must be remarked (as a general theorem which we shall use hereafter without further explanation) that an ex- pression of the form
E. sin —* (vt—2) + F. cos =" (vt—2)
may always be put under the form
JEFF. sin = (ve —2£ = : where tan G=5, and G is constant for that ray. It is plain that this expresses a periodical vibration similar to that which we have all along supposed, and whose coefficient instead of ¢ is ./ EF? +F*. It is convenient to take the square of this coefficient as the measure of the intensity of light: and thus E°+F” will represent the intensity in all cases similar to that before us.
In the present instance, the intensity or the sum of the squares of the multipliers of the sine and cosine of “ (vt — x) is
c feos? a + p.cos’ pd + sin’ a + P.sin’ p Qa _—— — . + 2eos— O.sina + P. cosa + p.sin P. cos pt
= pa + cos 2.a+ $.cos 2h + cos =" 6. sin 2.a + p.sin 2ht.
Thus we have a general expression for the intensity of the light when polarized light passes in any one direction through the crystal, and after being reflected by the analyzing plate, 1s recéived on a screen. If we suppose polarized light to fall in all possible directions (within certain limits) upon the crystal, we must give all possible values to @ and ¢, and we shall have the
Double Refraction of Quartz. 91
intensity of light on all the different parts of the screen. It will be remarked that @ is very nearly proportional to the radius vector of the corresponding point on the screen, and ¢ the angle measured from the lower part towards the right.
If the eye be placed to receive the light coming from dif- ferent parts of the analyzing plate, the appearance will be re- versed with regard to right and left, and here the angle ¢ must be measured from the lower point towards the left.
1*. Let the plates be crossed, or a=90°: cos 2.a+gm=—cos2p: sin2.a+pP=—sin 2g:
and the intensity is
2 <. sin’ 2p. sin® st
or putting for © its value, the intensity is
(ge wp fix @— ,, 3: sin 2p.sin —* a) This is 0, or there is darkness, if ¢ = 0, or = 90°, or=180°, or = 270°, whatever be the value of 6. This shews that there is a black cross through the center, parallel and perpendicular to the plane of reflection.
Also there is darkness whatever be the value of @, if
Tr #0 mr YD
.=0, =7, =2n, &e.,
2br 4br
. 2 a = or if 0 =0, a (iP ee b?)’ T (a —b*)’
&e. This shews that there is a dark spot at the center, and a succession of dark rings, of which the difference between the radii diminishes con-
tinually. Czteris paribus, the squares of the diameters of these rings are
, a—b : inversely as — 5’ oF are least in the crystal where the double refrac-
M2
92 Proressor Airy on the
tion is greatest; and are inversely as 7’ or are least when the thickness of the plate is greatest. They are directly as \, and consequently are greater for red rays than for blue. Hence after a little time the bright rings of one colour correspond with the dark of another. This gives the peculiar coloured character to the rings: it also prevents any of them from being totally black: whereas the evanescence of light in the cross is independent of d, and the cross is totally black.
2°. Let the plates be parallel or opposite: or a= 0. The expression
for the intensity becomes
ri}
5 §1 +c0s* 2 + 08 2 ©. sin’ 2g}. If ¢ = 0, or = 90°, or = 180°, or = 270°, this becomes c®: thus there is a bright cross instead of a dark one. or other values of @ the light is
TT, Ree cas greatest if aoe = 0, or = 27, &c., and least if Fas =r, or=3n, &e.:
in the former case it =c*, in the latter c’cos*2p. These indications point out exactly the form of fig. 2. In fact it is easily seen from the. expressions that the intensities in corresponding parts of fig. 1. and fig. 2. are precisely complemental.
3". In the general case, if sin 2p=0 (that is if @=0, or=90°, or =180°, or=270°) the expression becomes
r0| & 20| 9
{1+cos 2.a+.cos 2p} = — $1+cos Qa}.
This indicates a faint cross, which is bright when a is small, and dark when a is nearly = 90°. And if sin 2.a + = 0, another cross of equal intensity is found, inclined to the former at an angle a. Generally if
be between 0 and 90°— a, the intensity is greatest when =T@ = 0,
= 2, &c., and least when “2 O=7, =37, &e.: but if @ be between
90°—a and 90°, the intensity is greatest when =" 9 =7, =3n, &., and
Double Refraction of Quartz. 93
least when it=0, =27, &c.: and the same holds if we increase all these angles by 90°. Thus there is in fig. 3. a mixture of parts of the two
systems of rings in figs 1. and 2.
II. Now let a Fresnel’s rhomb be interposed, and let RAr, fig. 16. represent the plane perpendicular to the plane of internal reflection, and making the angle 6 with the plane of original
Jn the ety 2 polarization; the rest as before. The vibration c.sin <_ (vt — 2)
perpendicular to P,A may be resolved into
2 e.sin +" (vt—a).sin B parallel to Ar,
2 2 and e.sin (vt—2x).cosB perpendicular to Ar
Of these the former, by Fresnel’s theory, is retarded one quarter of an undulation, or the latter is accelerated as much. Adopting the latter supposition, we must suppose that after the emersion
from the rhomb, the vibrations are
c. sin = (ta). sin B parallel to Ar,
aed, Q and ¢.sin =~ (vt —2) + 90°. cos B, or ¢. cos ~~ (vt— 2’) .cos
perpendicular to Av. These will furnish for the ordinary ray of the crystal
2 Nae : = ; ¢-sin -— (vt—x). sin B.sin B+a+pt+c. cos = (vt—a).cos B.cosB+a+ and for the extraordinary ray
2 am =a 9 ————— —¢-sin5~(vt— 2). sin B.cosB +a+ p-+e.cos—~(vt—2). cos B.smB+a+¢.
94 Proressor Airy on the
After emerging from the crystal, we must (as before) diminish 2 in the latter expression by 9: and thus the vibration of the extraordinary ray
=-—c.sin | (vt—# + @). sin 8. cos Bt+at+o
+¢.c0s =" (vt — x + ©). cos B.sin B+ a +g.
The only parts of these transmitted by the analyzing plate are the resolved parts perpendicular to its plane of polarization = vi- bration of ordinary ray x cos @ + that of extraordimary ray x sin ~
=c.sin = (vt—z).sin B.sin B+a+.cos
Cee oe ee + ¢.cos—" (vt — x) .cos B.cosB + a + pcos p - Qr , —_————- . —e.sin — (vt—a+ 6). sin B. cosB+a+p.sin p Qar _ so aa c.cos =~ (vé—a + @) . cos B.sin B+a+q@. sin ©. The coefficient of sin = (t— 2) is c.sin B.sin BHatp. cos p—e.cos—" O.sin B.cos B+atg.sin Dr. ——— SS —¢.sin — 6. cos B.sinB+a+p.sin p: : Ir c the coefficient of cos a (vt—2x) is c.c0s B.cosB+a+p.cos p—c.sin—" O.sin B.cosB+atp.sin g
2 _ SS +c.cos—* @. cos B.sin B+a+q¢.sin .
Double Refraction of Quartz. 95
The intensity, or the sum of the squares of these coefficients, is (after reduction)
Sii+ cos 23.cos 2p.cos2.B+at+g Ir 5 F ——— . Aa ; , + cos —— @. cos 23. sin 2p.sin2.B+a+p— sin | 0. sin 28. sin 2p}.
Now if B=45°, cos28=0; sin2B=1; and the intensity = Sji-sin =o . sin 20.
1". Since a has disappeared from this expression, the figure will be the same whatever be the value of a, that is, whatever be the position of the analyzing plate.
2". When ?=0, =90°, = 180°, = 270°, the expression becomes =
which shews that there is a faint cross parallel and perpendicular to the plane of reflexion at the analyzing plate.
3". When ¢ is >0< 90°, or >180°< 270°, the intensity is greatest
Bae 3 aie 5 4 if Sharia e, &e. and least if 70 = = i soi &. When ¢ is >90" < 180°, or >270° < 360°, the intensity is least if — Q= = MEE he, -0 20 a On d — == —— : and greatest if = te) Ap ae &e
4". If 8=135°, the expression becomes c 5 . aa ; 2 Tig + sin > @.sin 2:
from which it is easily seen that the bright parts of the quadrantal rings in this case correspond to the faint ones when $=45°: and vice versa.
III. If in the last experiment the rhomb be placed in posi- tion 0, we must make 8=0, which gives for the intensity
e —— ON pe b een 3) + cos 2p.cos2.a+p-+ cos 7 ©: sin 2p.sin2.ar gh,
exactly as in the general case of experiment I.
96 Proressor Airy on the
If the rhomb be in position 90°, the expression is exactly the same.
If the analyzing and polarising plates be crossed, we must make a=90°, and the general expression becomes
2
S {1 -eos 28.cos2p.cos®. B+
— cos" ©. cos 28. sin 2p. sin 2.B+p— sin <7 6. sin 2B. sin 2g}.
2 Here if ¢=0, =90°, =180°, =270°, the intensity is = sin’ 2B, which shews that there is a faint cross parallel and perpendi-
cular to the plane of reflection. For other values of ¢ the only variable part is comprehended in the two last terms or
k : seas ‘ —— — = sin 2p. }sin 28.sin Ot cos 2. sin 2.B +p cos" Of.
This may be put under the form A.cos (= e- B),
where
A=—Ssin 29 sin’ 28 + cos*2.sin*2. B+, and tan B= ee oe
The equation to the dark rings will be found by making =" 9— B=0, or = Qn, or = 47, &e.;
hence 9 =*8, or =n 428, or = 20 +98, &e. Tv
and 6= V Tog VE oe V gg VIE, or = V Fe V renee &e.
Now when £ is small, sin 2.8+¢ being positive, tan 28 is small,
and tan B is small: except sin2.8+¢ is small, when B sud-
Double Refraction of Quartz. 97
denly becomes=90°: when sin 2.8+¢@ changes its sign, B changes its sign, and is = — 90°; its magnitude then diminishes till sin 2.8+p=—1: and it then goes through the same changes. The circle is therefore changed into the form represented in fig. 17. But the rings at the parts where sin 2.8+@=0 being very faint, the bends of the curve scarcely attract the attention, and the figure appears elliptical. But when £ increases, the intensity of the rings where sin 2.8+=0 is not small, (it is represented by 4), and the change of form is easily seen*. All these conclu- sions correspond perfectly with observation.
IV. To investigate in a similar manner the appearances pre- sented by plates of quartz, on the suppositions made in the begin- ning of this paper, we must resolve a plane-polarized} ray into two elliptically polarized rays. In fig. 18. let AP, be the plane of primitive polarization: AC the principal plane of the crystal ; the ordinary ray being elliptically polarized, will consist of one vibration in the direction Oo,, and another in the direction 0, 0, following it one quarter of an interval of undulations: the coefti- cient of the latter vibration being=s x that of the former, where k is a fraction depending by some unknown law on the incli- nation to the axis, but becoming=1 when the inclination=0, and =0 when the inclination is considerable. And the extraordinary ray will consist of one vibration in the direction Ee,, and another in ¢,¢, preceding it one quarter of an interval: the coefticient of
* I have investigated (in nearly the same manner) the form of the curves, supposing the crystal placed between the polarizing plate and the rhomb. The calculated pheno- mena are nearly the same as those described, and agree perfectly with observations.
+ I use this term instead of rectilinearly-polarized, the natural derivative from Fresnel’s substantive, only because it is shorter.
Vol. LV. Part I. N
98 Proressor Arry on the
the latter being = i x that of the former. Let the vibration per- pendicular to AP, be c.sin = (vt—x) or c.sin— (for brevity): and let the vibrations be In Oo,, p.sinE+v, or pcos v.sinE+p sin v.cos & or w sin £ +2 cos &. In 0,0,, kp.sin §+v—90°, or —kp.cos€+v, or kp sin v.sin §—kp cos v.cos &, or £x sin E—kweosé. In Ee,, q.sin F+yx, or g cos x.sin + g.sin x. cosé, or y.sin E +x. cos &. In ee, 2. sin F+x%+90°,
Fi : : q iis Y or 1 cos é+ x or — 4 sin y.sin E +r 7, C08 x - C08 E, or — 7, Sin + 700s &.
Resolving these in directions parallel and perpendicular to AP,,
and comparing them with the vibration from the original polari- zation,
cos a+ (w sin E+& cos t)+sin a+¢@ (kw sin &—kw cos £) +cosat+¢@ (ysinE+s cos£) + sina+@ (— z sine + cos €) =e sin &,
sin a+@ (w sin E+ cos t)—cosa+@ (ka sin &—kw cos é) +sin a+ (y sin E+8 cos £) — cos ato (— ; sin — + cos £) =0.
Or (since these equations ought to hold for all values of é) equating separately the coefficients of sin € and cos é,
i aie = e= —— sin &f TEP wth sin oF G.2 +e oF G-y — Ge He coevee (1),
cos atp.a—k snat+gd.w+cosatp.® + sete y=0
Double Refraction of Quartz. 99
sinat+$.w—k.cosatp.xtsinatg.y + S0*P, 9 noBtes (3), sinat+$.2+h.cosatg.w+sinatg.x —SSFPy_9 aproce (4).
By the solution of these equations
Cc —_— w = Tap esate,
oe é c.sina+ Toe Peery,
Y= Tye ©: 008 at+@,
| aero s=— +p °: sin at @. And hence the expressions for the vibrations are (omitting the
c common factor ee Se ear -_ —— Qa In 00, cosa+@.sin ZL (et 2) +h. sin at+¢. cos TL (et- 2).
In 0,0,, #. sin a+q@.sin =" (vt—x)—k.cos a+. cos = (vt—2).
—_
In Ee,, I. cos a+ p.sin = (vt—x)—k.sin a+.cos =" (ota).
In e,¢., sina+q@, sin <7 (vt—2)+h.cos a+. cos = (vt—z),
After emerging from the crystal, « must be diminished, in the two latter expressions, by 9. (0 is in fact negative for quartz, but that circumstance makes no difference in our investigations or conclusions),
N 2
100 Proressor Airy on the
Now when the light is received on the analyzing plate, the only parts sensible are those perpendicular to its plane of re-
flexion. They are, (putting — as before for <7 (wt —2)),
From Oo, cosa+.cosp.sinE+k.sina+.cos.cosé. From 0,0,, #.sina+@.sing@.sinE—k.cosat+¢p.sing.cosé.
From £e,, #. cosat¢. cos. sing + “70 —k. sina+¢. cos. cos E+ 220
=z
ae +k.cosa+.sind.cos’+ "=
From ee, sina+g.sing.sing +
Taking the sum, the coefficient of sin — is
Ia i ha ali
“7 Inr09 + k.sina.sin ea es: sin @. -COS——.
The coefficient of cos & is
ksina +k’. cosat+.cosp.sin =70 isin a .COS a0) sina +.sing,. sin = The sum of the squares of the coefficients is (after all re- ductions)
ae 8 Sanaa ce ae 2 (1-2)? cos'a+ 2g. sin? — if (i iad + 2k. sin a.sin™*) :
And restoring the multiplier a5 we have for the brightness
qd 7
sin a. sin — 7).
eGaen cos’.a + 2o.sin— + c° (co neers 17) Oe p. 7 ( 8a. CO8 >— + Ti
Double Refraction of Quartz. 101
When the plates are crossed, or a = 90°, this becomes
1 ay aA ocr \S) ; Ak? ie TO) 2 (9) - = ace = ep = c G aie -sin’ 2. sin aia +e a+ FP) sin’ > aE a0) i WS es = (is ~ —_ ¢ —_______ —— 9g , ee aN te + hk’) Ca TOTES SI cay of
1. For any value of @ this is 0 when
a = =0, =7, =2m, &e.
This shews that there are dark rings, exactly circular: it represents cor- rectly the experimental fact.
2. But since by our 4" hypothesis the spheroid and the sphere, used for determining the course of the two rays, do not touch, © will have some value when @ is 0, It cannot therefore be expressed simply by
ha
2b
Tx x 6,
but must have an additional term 7'x H. The value of E (as depending on A) may be thus found. At the center & is supposed (hypothesis 3) to =1. Consequently the general expression for the intensity of light at the
center is
A 79 5 . wO\* ; Fs 7rO c (cos «..cos = ate sin asin” ~) or c’.cos (« — =):
This is 0 when a — “e = 90°: or putting 0’, a’, for these particular values of © and a, a =90 + = Now it was found by M. Biot that in a right-handed crystal, a’ (measured in the direction that we have supposed) must exceed 90° by a quantity proportional to the thickness of the plate directly and the square of \ inversely. That is,
ut LL
eT ne one Nerd
I hig os
102 Proressor Airy on the
whe e a-b EF. therefore is 3 and consequently 6 = 7'x (— + eRe 0 y:
We may remind the reader (as we shall have occasion to use it afterwards)
that eh is the angle through which the analyzing plate must be turned
from the crossed position to produce darkness at the center for the particu- lar colour used. We may also remark that, if we still consider © as posi- tive (which we shall continue to do), all our expressions (as appears from this comparison of theory and observation) must be understood to apply to a right-handed crystal: if the sign of & be changed, they will apply to a left-handed crystal. But if, more correctly, we put a negative symbol for ©, then our expressions would apply to a left-handed crystal, unless the sign of & were changed.*
Oo. Te Tr(@-B6),. The value of + 8 therefore et a Ge
that \’ + 0A may represent the length for rays of any colour, \’ being that
Let us suppose
for one of the mean rays, and therefore constant, and dA being small for
: ; 9 all the bright colours. Then for ara aa? We have
Te , Tr(ad - 38’) 9 orate | Tr(a’ — 6) e
x2 en) ar xy a ay aa ) nearly. The mixture of colours in the rings will depend only on the difference of
the values of 7° for different colours, and not on its absolute values, and
* The reader who will take the trouble of tracing the expressions will find that, if the sign of © and of & be changed at the same time, not only will the right-handed-ness of the erystal remain the same, on comparing the expression with Biot’s experiment, but also all the directions of the spirals &c. in the succeeding experiments will remain the same. Thus the connection between the right-handed-ness and the direction of the spirals is independent of the sign assumed for ©. With this consideration, I have thought it best to use the same symbol in the theorems for cale spar and for quartz.
Double Refraction of Quartz. 103
consequently only on the last term of this expression.* Now this is the
same that we should have had for the colours in Newton’s scale depending
: 2T T x(a? — b° on the thickness Vv en ae 7 ) 6 of a plate of air, if
2
a—b
ab be the
2 72 same for all colours: or if the variations of — be proportional to the
variations of ), the last term must be altered in a certain proportion.
and our statement will still be true. It appears therefore that the central
tint will be nearly that corresponding in Newton’s scale to a plate of air whose thickness is fa : and will be followed by the other orders of New-
ton’s scale. This agrees sufficiently with the observations.
8 : 3 3. Unless =A = 0, or =7, &e. the light is not = 0: therefore there
is no black cross. But the light is least when ¢=0, or = 90°, or =180°. or = 270°: and greatest when @ = 45°, or = 135°, or = 225°, or = 315°. This shews that there are dark brushes parallel and perpendicular to the plane of reflexion, but not interrupting the rings. As & is nearly = 1 in the neighbourhood of the axis (by hypothesis) they are not sensible near the center: but as, on removing to a distance, k approaches to 1, they
become stronger. This is conformable to observation.
* This is not strictly true: for though, in comparing this with one of Newton’s rings, the colours mixed may be the same, their intensities will not be equal except sin? = be the same, and consequently the compound colour will not be the same. But it is plain that, if
we take the ring preceding and that followiug the point where the difference of ae is the
} 2 : 9 : same as in the case before us, then at the points where sin? — has the same value for the
mean rays, we shall have mixtures differing in opposite ways from that under consideration. In the same manner, in the rest of the paragraph, it must be understood that, if we take those colours of Newton’s scale between which the colours of these rings lie, we shall always advance in the scale: but we may possibly have no ring intermediate between two of New- ton’s, or we may possibly have more than one.
104 Proressor Arry on the
V. Taking the general expression for the brightness, and
: Qk making tan y = i+ F tan a, we have Praia) + as sina. sin = V cos?a + ae sin* a. cos es A Seal kr TI a. — = on ‘ palais paige Oo hot x SST a haa pag aa aid
and the general expression for the brightness is
RE RECN DN ee. 8 iO Bins © (Fp) - 00s" a+ 2G. sin* = +e (cos! a + Gap sina) . cos! = —
Supposing that is not much altered by a small alteration of 0, this is a maximum or minimum for a given value of ¢ if
>
—— . 2r9 S12 : f 3 0= (1-2). costa + 2p .sin ~~ (I+ Ff. cos’ a + 4 sin®a). sin27? oy,
279 or tan
= tan a x Lt Pe costa + ah sin’a + 1 — # ° costa + 2h 1+ 1+#} .cos’a + 4H sin?'a —1 — kh}. cos?a + 2h
Therefore an8 will be greater than ¥ (or Y +7, or ¥ +27, or
+37, &c. for each of these satisties the equation tan yaaa tan a)
by the angle » whose tangent is the second side of the expression.
1. If & be nearly equal to 1 (which we suppose to be true when @ is small) and a less than 90°, the expression for tan w is always positive :
its greatest value, when
¢ =90°— =, or 180°— >, or a1 = or 360° —'<-
2 " 9, A =o 212 ene is Bang ae 1 Brine, 1+ 4k? —1 — #|?. sin®a
: gone on De a! and its least, when p = 45 3° 8 135 ? OF 225-5, or 315° — 5, 2k
1S l+F tan a.
Double Refraction of Quartz. 105
279 r
+7, or Y + 27, &. by an angle w which is always included between 0
Therefore in the bright or dark rings
will be greater than y, or
and 90°, and which has its maximum at a part which, when a is less than 90°, is found by looking a little to the negative side of the perpendicular and parallel to the plane of reflexion (thus, if the crystal be right handed, we must look to the left of the upper part, and the points of the square appearance will be found in that place). When a is greater than 90°, the maximum value of the tangent takes place for points found by looking a little to the positive side of the perpendicular and parallel: but the tangent is then negative (for tan a which enters as a multiplier is negative). Con- sequently the maximum contraction of the circle is found by looking to the positive side, and the points of the squares will be found by looking to the negative side. Whichever therefore be the direction in which the analyzing plate is turned, the circles will be changed into the form represented in fig. 15 (the crystal being supposed right handed). This remarkable conclu- sion agrees perfectly with the facts of observation.
2. If % is very small, the expression for tan w may become negative, which shews that w will suddenly exceed 90°, and after having continued so during the change of @ through an are of various extent (according to the value of a) will suddenly become less than 90°. This shews that the form of the bright and dark rings will be that of fig. 2, except that in- stead of absolute interruption of the rings by the eight radii, they will pass very highly inclined through those radii. The rings of quartz become so faint at a distance from the center that I have not been able to observe
whether this is or is not supported by fact.
3. We have already noticed that when a = 90° + ce there is a dark
spot at the center. Now for any given value of 0, it appears (from the Vol. IV. Part I. O
106 Proressor Airy on the
general expression) that the light is least when
p = 45° — + 135° — =, &e.
This shews that, with this value of a, the spot will be a darkish cross, its
arms in positions
a ote 45 — 3? 135 3° &e.
But these are exactly the angles at which the depression of the ring below a circle is a minimum, or at which the points of the square are found. Therefore, we may expect to find a cross-like spot in the center, its arms in the diagonals of the square. This corresponds perfectly with the phe-
nomena.
4. The succession of colours in the cross-like spot, it is easily seen, depends only on this circumstance: that as A is greater for red than for blue rays, the value of a which allows no red rays to pass is less than that which allows no blue rays to pass. That is, for a certain small value of in there are rays of the blue and only of the spectrum transmitted: for a
larger value, the rays only of the red end are transmitted.
5. If the polarizing and analyzing plates are parallel, or a= 0, the
expression for the brightness becomes
2 2 +e @ sho 72 ce -— 31 tee - COS 2p} sin a °
It is easily seen that this indicates a series of rings, but there is now no total darkness. When
¢=0, =90°, = 180°, = 270°, the expression is
5 6 4h? 79 C 2
eae xo
Double Refraction of Quartz. 107
As & becomes smaller, this varies less with the variations of 0, but does not vanish: that is, in receding from the center, the rings are more and more interrupted by a white cross. If the plate of quartz be thin, it may happen that the first ring is so large as to be sensibly interrupted. In this case (as the first ring is broad) it will lose the appearance of four in- terrupted quadrants and become four dots. This is easily seen in experi- ment.
VI. We have seen that, for a bright or dark ring, 278 will
differ from ¥~ only by » When a is 0, » is 0: and when a is 90°, » is 90°, having increased gradually to that value. Also when a is 0, ¥ is O or z, or 27, &c.: and as a increases gradually to 90°, y increases gradually to
Tv Se oi Ob cas &e.
27r0 4 F : Consequently, = or ¥ + increases by 180°, while a increases
from 0 to 90°. In the same manner, as « increases gradually from 90° to 180°, » increases gradually from 90° to 180°, and y from
7 3a 5a g° or og? OF ae &e. to 7, or 27, or 37, &e.:
2790 Toe and consequently — or ¥ + again increases gradually by 180°.
The same holds for every successive quadrant of revolution. Thus, if we fix our attention on any ring, and turn the analyzing plate to the left (the crystal being right-handed) the ring’s distance from the center (or 6, which
‘ / 2b Te TG m\°-=))
a
will increase continually, but not uniformly. This is conformable
to fact. 02
108 Proressor Airy on the
VII. When Fresnel’s rhomb is interposed in position 45°, we must suppose AR, fig. 18, to be perpendicular to the plane of internal reflection, and we shall have the vibrations
= ae oe (ve —x) parallel to AR, and Fi es = Ate — 2)
wee to AR. Resolving these in directions Be and perpendicular to 4C, we find for the former
= Fa f sin2” (ota). cosa P+ 45+ Fe - COS = (vt—2).sinat pt 45 or — =F sin =" (ot — 2) — (a + 9) — 45°: and for the latter, iF sin 27 (vt—2).sin opti Te cos = (vt—2) .cosatp+45", or 7 cos 27 (wt — x) — (a+ ¢) — 45°.
Now taking the same expressions as before, for the vibrations in Oo,, &c. and comparing the sums in directions parallel and
perpendicular to AC, (putting 27 (wt —x) — 45°=£), we have “sin F—( +) or wie E+ ee ——s —\a as! a Fa Ja a+. sin sy sina +o. cos & s\. = (ke - ) sin — + (3 — kw) cos &
c Cia c —_— Wik a ~—(a+) or ie ee geen er? cme
= (w + y) sin & + (x + 8) cosé
Double Refraction of Quartz. 109
Comparing the coefficients of sin ~ and of cos £,
bs) Cc ———
kz — i x ag ae
thw = Fy nat @ OQ %
wry = ge LP
tb —_——
ee RB ere
Obtaining the values of w, x, y, x, and substituting in the expres- sions for the vibrations, we find (omitting the common factor
aam a) (1+ 4h) /9/° In Oo, (1-4). cosE- (a+). In 0,0, &£(1—h).sin E-(a+¢). In Ee, £(1+8) cos —(a +). In ee, —(1+h).sin F—(a+ 9). After emerging from the crystal, ¢ in the two latter expressions
2 = must be increased by ——.
Taking the neath parts of the vibration perpendicular to AP., we find for the efficient vibration,
(1—-). cos E—(a+@) . cos p+k(1—k).sin €—(a+ @). sin p
+h (1+h).cos&—(a+¢) + 270 cos p—(1+h).sinE—(a+¢) + 270 sin op. The coefficient of sin —(a+¢) is
k(1—k). sin p—k(1+h).cos @. sin” To (142). sin Pp. cos
hay A
110 Proressor Airy on the
The coefficient of cos —(a+¢) is
(1—£).cosp@+ (1+4).cos p. cos “20 — (1+4).sin ¢. gin The sum of the squares is
1+F] -2k.1—#. cos 26+ 2h.1—F*. cos 2p. yee
—(1-').sin 2g. sin =
r
Restoring the factor ae niet we have for the brightness cf 2k.(1—-#) ia il
2k (1—k*) 270 Mey ues 2p + +P) cos 2. cos —
r
_ p22
. . 2rO eure ae x \.
; +k? If we make tan x = aE E) ok 14+ (1+
tan 29, the two last terms, o1
Ems cos 2. cos = — sin 2@.s1n sae
a $ a cos’ 2p+sin* 2p. cos pe es and the expression for the brightness is
c 1-k 4k 2h
= iI ~The va cos’ 2p-+ sin* Wh -+ [ze Cos *#)
aS 1-f Ake 1+F (1+4*)°
become LEE 1+
cos’ 2p+sin’ 2H. cos? ~ 2 +% 1;
"i
As the multiplier of cos*—— +% i
9 is never = 0 while # has any value between 0 and 1, the rings are not interrupted in any part of their circumference.
Double Refraction of Quartz. 111
2”. This multiplier however is small when & is nearly=1. It is also small when £# is small, if sin 2=0, that is, when P=0, =90", =180°, =270°.. Hence there will be no rings very near the center: and at a distance from the center, they will be faint in the lines parallel and perpendicular to the plane of reflexion.
3". The form of the dark rings will be determined by making
cos? 2° + X=0: and that of the bright ones by making it=1. The first of these suppositions gives
ae) 9e ar lar é iss) a 9e
eg igs Sag Se andi tg 9? Ne.
Now x increases from 0 to 90°, from 90° to 180°, &c., while 2 increases from 0 to 90°, from 90° to 180°, &e.: consequently x never differs much from 2: and therefore for the dark rings
tO & 30 ne = 9 _ Pp; ie — Oo _ Pp, &e. nearly.
That is © increases continually as @ diminishes: and consequently @ in- creases continually as @ diminishes. This shews that the curve is a spiral, and that (reckoning from the central fold) it is turned in a negative di- rection; the eye when fixed on a part above the center must turn to the left to trace the curve as it recedes from the center, supposing the crystal right-handed. If the crystal be left-handed, the sign of # must be changed : this changes the sign of x, and the spiral is turned in the opposite di- rection. This agrees perfectly with observation.
4. If we take the radius vector in the direction opposite to that cor- responding to any point of the spiral, that is, if we increase p by 180", we find for the new values of © in the dark rings 30
= ———— 5 ~ +180, — P4180, — G+ 180, &e,,
via 30
TT f . or sah iets eae rm’ &e. nearly.
112 Proressor Airy on the
This is exactly the same series as that for the original direction of the radius vector: and therefore the values of @ are exactly the same for any radius vector as for that opposite. But the curve (as we have seen above) is spiral. These two conditions require that the form of the dark line be two similar spirals mutually inwrapping each other, their positions differing by 180°, But no other alteration of @ will give the same values of © in the dark rings. Consequently the form of the dark rings is two spirals, and only two, turning in the same direction, and in opposite
positions. This remarkable conclusion is supported by fact.
5". When 2¢ is between 0 and 90°, yx is greater than 2: when 2 is between 90 and 180°, yx is less than 2p: and so for successive quadrants. That is, when @ is between 0 and 45°, y is too great, or 8 too small, for a spiral of uniform approach to the center: when ¢@ is between 45° and 90°, © is too great. This shews that the spiral will have a square appearance, the right-hand angles being higher than the center. This is precisely the form really presented to the eye.
6". The expression for the brightness of the center (where #=1) is
j é Bie atis z simply a: As this is independent of \, it shews that there is the same
mixture of colours at the center as in the light which we use: and
that therefore with common light the center is white.
I should only take up the reader’s time unnecessarily by going through the investigation with the rhomb in position 139°.
VIII. We have found (in the investigation of IV.) that put- ° Qn ‘ P . as ting & for <_ (vt-2), the expressions for the vibrations when light, at first plane-polarized, has passed through a plate of rght-
handed quartz, are (omitting the common multiplier i+ a)
Double Refraction of Quartz. 113 In O00,, cosat@.sin E+h.sin a+. cos &. In 0,0,, sina+q.sin €—k.cosa+@. cos E.
- 270 La) In Ee,, cosa+¢. sin E 4 =79 _ sin ato. cos — + Ney:
._— . —— 270 In e,e., sin at p.sin E+ =7O 4 heos a+. cos E + ——.
We have now to resolve these into the elliptical vibrations in a plate of left-handed quartz of the same thickness, with its principal plane in the same position. It will be observed, that the difference of right-handed and left-handed quartz consists only in the difference of the sign of 4.
Taking first the vibrations in Oo, and 0,0,, and using those letters to express the vibrations in those directions,
Oo,=vibration in O’o', + that in E’e’ =w sin +x cos &+y sin +2 cos &,
0,0,=vibration in o',o', + that in ee’, =—hkax sin E+hw cos — + ; sin & — 4 cos &.
Comparing the coefticients of sin — and cos é,
a= 1-F cos at+p=wrty, w= 1 Dy aes at¢, ksina+p=2 +x, ALS at, 1+ Hsina+o=—hka +=,¢ Whence es, a Y= 14 pcos at, —k csatp= hw—¥ a : ke ® sin atd¢. ~1+k
Vol. IV. Part I. P
114 Proressor Airy on the
Whence the vibrations, after emerging from the second plate, are (omitting the common multiplier tae addition to the former) :
In O',0; 1—#.cosa+.sin & + k(1—#’).sin a + p. cos &.
In oo, —H(-#).sinat p.sin & + k(1—F). cosa + p. cos &.
= . ea 9 In Ee’, 2k. cosa +. sin +920 + 2h’. sin a + p.cos— + are. ‘ee a 270 ——, 2rO nrere. 2H. sin a+ p-sin E + —— — 2k cosa + .cosE + 27° Similarly for the vibrations in Le, and @,e,: ; 270 270 He, = wsint +——+ x cost + —— r r d 270 27rO +.y sin &.4+.—— + 8003 £ +———- r r , _ », 200 ~ 20 é,e@. =—hkx sin & + ane + kw cosé + a2 Ss 270 y 279 Fe AU ere gee es : ages : 2790 Comparing the coefficients of sin — + =n8 and cosé + a @ ee ee — 2 K cosatp=wty, w= pet, —k si = ae sts Epes t= sin a+, i a=. * \ whence pe oe Teta | tae _-FO-*) way9 Fae op tet Be : k cosat+p= kw — 5, 2 hk x a6.
' 1+h
Double Refraction of Quartz. 115 Whence the vibrations, after emerging from the second plate, ae 1 are (omitting TB): 270
In Oo, 2h. cosa+.sin F + = —2h.sina+@.cos pis 250 In o',0'. 2k. sina+@.sin E+ =e +2 cosat+@.cos p44 278 In E’e’, —2(1—F*) cos a+¢.sin = . +k(1—#*) sin a+. cos — + —_. In ee. (1—4*). sin a+q@.sin — + ——— +k(1—#) cosa+g.cosé + si, Adding together the vibrations in the same direction, we find
Ai c (omitting the factor ae aoe
Vibration parallel to Qo, =1—#'.cosa+o.sinE +h. (1 - kh) sina + p.cos &
SSL OTT er ees 2rO + MMP. cos.a+ p.sin E+ 7° ~ 2k (1 ~ #) ‘sina + p. cos E + —
—# (1-H) cos + p.sinE +929 +k(1-2). sina +p .cosé +979,
PQ
116 Proressor Airy on the
Vibration parallel to 0,0,=
—k#(1—#).sina+¢. sinE + k(1—#).cosa+ p.cosé
+48 sina tp. sin€ +222 — 9h(1— 2) coset G- cost + =T—
4 (1—#). sina + p-sin E+ 222 + RF) cosa p. cosk + 2°.
To avoid unnecessary generalities we will suppose the plates crossed, or a= 90°: which gives Vibrations parallel to Oo, = —(1—#). sing. sinE + k(1—#). cos p . cosE — 4 sing.siné + 2x0 — 2k(1—F) cos p.cosé + =n
+ (1 — ) sing. sing + S224 2-H) cos cos + “T~.
Vibrations parallel to 0,0, = —k(1— #)cosp.siné — k(1 — #’). sin p .cosé
2 + 48 cos .sinE + °=" + Qk — If) sin g cos E +22
+ (1-H) cos p. sine + ar0-k(1 — i) sin p. cosE + a8,
The efficient vibration, or that perpendicular to 4P., will be found by multiplying the former of these by cos ¢, the latter by sin ¢, and taking their sum. Thus we have
1-#
2 sin 2p .sinE + k(1 — #) cos 2p. cosé
Double Refraction of Quartz. — 2k(1— k)cos2.cos& + are
+— sin 2p. sing + 22° 4 £(1—#) cos 29. cos + ==— The coefficient of sin & is LS 2 F : - sn 2g + 2k(1 — #) cos 2h. sin ais 4 - : 4 ie Hsin 2p cos @™° — (1 — 2) cos 29. sin kaos The coefficient of cos ~ is k(1 — #) eos 2 — 2k(1 — F) c08 2. cos ===
4790
ne + k(1 — k’) cos 2. cos
le" Jis oe : + a : sin 2@.sin
The sum of their squares is, after all reductions, 1 PF sin? 2. {4k. cos ag. sin? — 2.(1 + #°).sin 2p. cos@ Ol.
Ga : ———.., we have for the brightness
And restoring the factor cea L—F\2 rm Ol AE a aks, “ 7 2 a —— —_ = . . ral ct. (Fs) - sin x SE eos 2. sin x 2.sin 2. cos If we make tan x= 1+ tan 2¢, this expression bec we make tan x= —57— tan 2, this expression becomes
ile e. TR (16H cos* 2p + 4.1 + Af. sin’ 2¢). sin’ 7° sin’ (
This vanishes, or there are black lines, when sin? — = 0, or when
(3) J ==Q; or =7, or =27, &c.
r
118 Proressor Airy on the
This indicates a series of dark circles, whose diameters are the same as
those of the circles seen with either of the plates singly.
2. The expression also vanishes when
or when
[=X =rtxy =Ar+yX &e.
Now when 2¢@ increases from 0 to 90°, from 90° to 180°, &c. x also in- creases from 0 to 90°, from 90° to 180°, &c.: consequently y will never differ much from 2. So that the expression vanishes when
Toe 29, =7+2¢, =2r+2¢, &c. nearly.
In the curve defined by this equation, it is plain that © increases con- tinually as @ increases, and consequently @ increases continually as @ in- creases. The curve therefore is a spiral in such a position that if we look at a point above the center, and follow the curve towards the right-hand, the radius vector continually increases.
3. Now if we increase @ by 90°, or 180°, or 270°, we get the same
values for x, increased by x, or 27, or 3. Consequently the values of the radius vector, at a point of the dark line, are found by making = equal in the first of these to m+y, 2r+y, 3rt+y, &e. in the second to 2a +x, 8rty, 4r+yx, &e. in the third to 3r+yx, 4r+yx, 5a+y, &e. These are evidently the same series of values as that found with the origi-
nal value of @. That is, if we draw four radii vectores of equal length at
angles of 90°, and if one of these terminate in a point of the dark curve,
Double Refraction of Quartz. 119
the others will also terminate in points of the dark curve. This condition can be reconciled with the spiral form of the curve, only by supposing that the curve consists of four similar spirals, each of which is turned 90° from the position of that adjacent to it. The general form of the curves will be therefore four spirals, in positions differing by 90°, and all turned the same way, intersecting a series of circles. This very remarkable form is
precisely the form given by observation. 4. The intersection of the spirals and circles is found by making both the equations sin® ze = 0 and sin? (72 = x) = 0, to hold at the same time. This gives xX=90, or =7, or =27, &e:
and consequently 3a
Tv gp =0, or = 9° or =7, or = 2° That is, the intersections of the spirals and circles will all lie in the lines through the center parallel and perpendicular to the plane of reflexion.
This is exactly true in the experiment. : : : s _ 79 . : 5. And since the successive circles require values of Suecessively in- creasing by 7, and the successive points of the spirals on the same_ radius - xO : : : ? vector also require values of “y ~‘Successively increasing by =, every circle
will be intersected by the spirals in the lines above mentioned, and there- fore every circle will be intersected at every quadrant. This is verified by experiment.
6. If we consider the parts near the center and not in the circumference
of one of the circles, the angle ¢’ corresponding to a dark point will be nearly
7 0’ 7 0 é Qn” or @, 790; &e.
120 Proressor Airy on the
where 9’ is the value of © corresponding to @ = 0,
pare or O= ath Consequently, eT eT A p FESS eee F & co) x? F oe + 90 c
But Ss is the angle through which the analyzing plate must be turned to the left to see the dark spot with the right-handed plate alone. Con-
sequently the dark cross in which the spirals originate is inclined, (the upper part to the right), by an angle half as great as the angle through which the analyzing plate must be turned (the upper part to the left) to see the dark spot with the right-handed plate only. This appears to
agree with experiment.
7. In the whole of this we have supposed the right-handed plate to be nearest to the polarizing plate. If the combination be turned, with the left-handed plate nearest to the polarizing plate, we must change the sign of &. This changes the sign of y (that of @ being supposed the same); and it will very easily be seen that in consequence of this change, the direction and position of the spirals will be exactly inverted. This is found to be true.
8. When & is small, the expression for the brightness becomes small if sin’ 2@=0, that is, if ¢=0, or = 90°, &c. This accounts for the dark brushes seen parallel and perpendicular to the plane of reflexion at a great distance from the center.
9. When ¢ is > 0 < 45°, x is > 2p: when @ is > 45° <. 90°, x is <2. Observing that the spirals intersect the circles when p=0, =90", &c., it is easily seen from this that they cut them at an angle rather greater than the angle at which a uniform spiral would cut them. This
seems to be observable in the experiment.
Double Refraction of Quartz. 121
10. The peculiar vividness of the colours appears to be explained by this consideration. The dark curves are defined by making the expres- sion for the brightness to vanish totally: whereas in some other cases (as in investigation VII.) the dark curves are defined by making the expres- sion for the brightness a minimum. Here then the colours are much less diluted with undestroyed white light than in investigation VII. And in the lines parallel and perpendicular to the plane of reflexion, y=0, and
: . ee eye oes) ’ the expression for the brightness is ¢ (14m 16H. sin Here then
is no sensible light for a considerable distance nearer to and farther from ; 5 iS) the center than the point corresponding to = na: the colours are less
mixed, and are therefore more vivid than perhaps in any other phenomenon of polarization. And the greatest brightness in these lines, if & be not very small, is four times as great as the greatest brightness in the lines making angles of 45° with them (as will be seen on making @=45" in the expression for the brightness). Experimentally the colours are brightest
in the lines parallel and perpendicular to the plane of reflexion.
It is almost unnecessary to point out to the reader that none of the peculiarities of these appearances would exist if & were either 0 or 1, that is, if the light were either plane-polarized or circularly-polarized. None of the expressions however would be
altered, if, instead of & we put i: that is, it is indifferent which
ray we suppose to have the major axis of its ellipse parallel to the principal plane. From the agreement between the observed and the calculated appearances, I think there is little doubt that the nature of the Vol. IV. Part I. Q
122 Proressor Airy on the
light in the two rays of quartz is such as I have described. I do not mean to exclude the possibility of supposing that the form of neither wave (in the construction for determining the course of the rays) is exactly spherical or exactly spheroidal: provided the difference of the forms be nearly the same as that of a sphere and a spheroid. Nor do I mean to assert that each elliptically-polarized ray consists exactly of two plane-polarized rays following each other at the mterval of one-fourth of an un- dulation: or that the ratio of the two axes in the two rays is exactly the same. But I conceive it to be perfectly certain that the general character of the light is such as is stated in my hy- potheses.
I have not made any calculations upon other suppositions, but I can hardly imagine that any other would represent the phenomena to such extreme accuracy. I am not so much struck with the accounting for the continued dilatation of circles, and the general representation of the form of spirals, as with the explanation of the minute deviations from symmetry, as when circles become almost square, and crosses are inclined to the plane of polarization. And I believe that any one who shall follow my investigations and imitate my experiments, will be surprized at their perfect agreement.
There is one relation between the construction for determin- ing the course of the rays, and the nature of the rays, which deserves (I think) particular attention. It is that (comparing the rays of quartz with those of any other crystal) a change in the nature of the ray is accompanied with an interruption of continuity. The nappes of the wave surfaces are absolutely se- parated. This is not the case in the common construction for uniaxal crystals, nor in Fresnel’s construction for biaxal crystals.
Double Refraction of Quartz. 123
There may possibly be a connection of the same kind as that between the change from partial reflexion to total reflexion within glass, and the accompanying change from plane polarized light to elliptically-polarized light. The cases are at least thus far analogous, that the change in the light and the interruption of continuity go together. But we are so much in the dark re- specting the physical constitution of quartz, that we cannot at present go farther.
It might have been desirable to verify my suppositions by more direct experiments on the separate rays of quartz. I can only plead that the duties of my office have not allowed me the necessary time. They would (under all circumstances) have been much more troublesome than those which I have the honor of laying before the Society: and I do not think that they would have been more satisfactory. The appearances presented by depolarization are admirably adapted to the discovery of the most delicate differences in the nature and course of rays. The same want of time I hope will be allowed as an excuse for the want of accurate measures*: without which no theory, however
satisfactory in general explanations, can be considered as firmly established.
G.B. AIRY.
OxpseRVATORY, CAMBRIDGE, Dec. 30, 1830.
Soc Lo ee a ee ee a eee eee
* It is much to be wished that the rings and spirals exhibited by quartz may be accurately measured, their diameters in different directions ascertained, and a comparison with theory instituted, by means of Biot’s measures of the doubly refractive energy of quartz. Should the observations and the theory disagree, it would shew, either that there is some latent error in the theory, or that the difference of curvature of the sphere and spheroid near their vertices is not the same as that which is inferred from Huyghen’s construction, modified as above
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IV. On the Resolution of Algebraccal Equations.
By R. MURPHY, B.A.
FELLOW OF CAIUS COLLEGE; AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read March 7, 1831.]
INTRODUCTION.
Tue researches of Lagrange on that part of Pure Analysis, which forms the subject of the present Memoir, have been fol- lowed up with considerable success by many foreign Mathe- maticians, amongst whom M. Augustin Cauchy deserves to be particularly distinguished ; indeed the extensive use of the theorem of Lagrange in Physical Astronomy, had turned the attention of Analysts to consider more intimately the nature of that series, the conditions of its convergence, the root which it particularly represents, &c. I have referred to as many papers on this subject, scattered through the Memoirs of the French Institute, the Journal of the Polytechnic School, and the Annales de Mathematiques, &c. as I conveniently could. I do not find that in the point of view in which this subject is here exhibited, I have been anticipated in any of the articles above referred to, a point on which it is necessary to be doubtful, without actual reference, from the great number of persons who have been recently, and are at present, engaged in extending the limits of Analysis.
126 Mr. Murpuy on the
I have supposed the given equation to be such as to contain neither negative nor fractional powers of the unknown quantity 2 ; if such should enter any proposed equation as ¢(«)=0, we need only put «=a+s, and consider = as the unknown quantity, since @(a+z) may always be expanded according to the positive and integer powers of x, when no particular value is assigned to 4a, this is therefore to be understood, unless where the contrary is expressed.
Suppose the root of an equation ¢(x)=0 is sought, the fol- lowing simple rule which I have proved and applied in Section 1. will give it with great facility.
“ Divide the given equation by 2, take the Nap. log. of the quotient by means of the formula
take the coefficient of the first negative power of x in this loga- rithmic expansion: this, with its sign changed, is the root of the proposed equation.”
If the proposed equation were of x dimensions, it has x roots, and it is natural to enquire which root is given by the preceding method. I have in the same Section shewn that it analytically gives a result which comprehends all the roots, but that arith- metically it gives the least root.
If, instead of the root of an equation, any function /(x) of the root should be required, there is given in Section (2) for this purpose, a rule nearly as simple as the above; namely,
‘Take the same Naperian log. as before. Multiply it by the derived function ,f’(z), the coefficient of the first negative
Resolution of Algebraical Equations. 127
power of x, with its sign changed, will be the required function of the root; minus the same function of 0.”
In Section 3, there is given a method equally simple with the former ones, to obtain the sum of any specified number of the roots; also the sum of any given function of a specified number of the roots.
In this Section it is also shewn how to find the m' least root of an equation, or any function of it, to which are annexed some remarks on that relation of imaginary quantities, which cor- responds to the relation of greater and less in real quantities.
As the sum of m roots exceeds that of m-—1 by one root, it is clear that by this method we can get all the roots of the equation, as well as any function of any root.
The principles laid down in the first three Sections are ap- plied in the fourth to the deduction of several theorems of ana- lysis; the theorems of Laplace and Lagrange are simple and almost immediate consequences: a theorem somewhat similar to Lagrange’s, which M. Cauchy gives in (Vol. IX. Memoirs of the Institute,) for the sum of any function of all the roots of an equation, I have here shewn holds true for any specified number of the roots as for the whole, though the series only terminates in the latter case, and under particular conditions.
There is also a theorem given by Burmann, which is of great use in transforming series, but the ordinary demonstration is very long, and may be seen in the notes at the end of Vol. III. Lacroix, Diff. Calc. but in the present method it follows in a few lines.
When we revert a proposed series, or find the root of an algebraic equation in a series, the law of the latter is not
128 Mr. Murpuy on the
readily visible: the same Section contains the expression of this law.
There is besides in this Section, a remarkable expression for the correction to be applied to the root of an equation, when to the equation itself there is added a small term.
Next in order, the important subject of the Limits of La- grange’s series is considered.
Section V. treats on Definite Integrals: in this Section the root of any proposed equation, any function of the root, &c., are all represented by means of Definite Integrals; and we here see the Analytical use of the multiplicity of the values of certain tran- scendent functions, when the quantity under the sign of such a function should be the same at both the limits of an integral.
The last Section contains various points of Analysis, connected with the present subject, which could not conveniently be brought imto any of the others.
Resolution of Algebraical Equations. 129
SECTION I.
To find the root of any equation ¢(«)=0, which contains only positive and integer powers of «.
Divide the equation by x, take the Nap. log. of the quotient ; the coefficient of the first negative power of x, with its sign changed, is the root of the equation.
For example, in the quadratic equation
a’ +ax+b=0, divide by z, and take the log. of the quotient, i.e. 1. (a+0+%) ,orla+l. (ogists)
by the preceding rule the root should be therefore the coefticient
1h of - in x
and selecting the coefficient of : which enters only in the 1",
3°, 5", &e. terms, we get for the required root
2,848 6.5 8.7.68 a) a eo 8 eg 8a Ra eS
as it evidently is, since this series represents the expansion of
- 6).
Vol. IV. Part I. R
130 Mr. Mureny on the
This instance of the application of the present method is sufficient to shew its nature, but as we may frequently facilitate the operation by a slight previous transformation of the proposed
equation, I have added a few more examples. To find x in the equation az’ +%s—6=0, in this case, the rule may be directly applied; but the result may be obtained, rather more simply, by putting s-—b=.. The equation becomes a(a+b)"+2=0, therefore by the rule
x= coeflicient of * in —1. {1 + “(a +b)"}
* 11 a a a a = = coeflicient of = in — ~, (« +5) + 4.9 (wt by" — 4. . (a +b)" + Be.
3n.3n — y OED yay So
2 = —ab" an be Bi cs 2 1.2
As another example, suppose a+z=1. (x) to find x, or s—e’.«*=0. Hence by the rule
Ate 1 es z = coeflicient of = in —1. (1-«.5)
Resolution of Algebraical Equations. 131
When the given equation contains fractional or negative powers of z, put (as was before observed) « = = +a, and considering * as the unknown quantity, we may directly apply the method.
The rule given in this section may be thus proved. Suppose @ (x) to be resolved into its simple factors; i. e.
(a) =C.(@—a). (w@—) .(«-y)
then £2) =c(1-$) (1-4) (1-2)...
where C’= C.(—£).(—y). &t. LPM 1c (1-$) +. (1-8) +1 (0-2) + &e
of which the only term which contains negative powers of x, is
p (a)
the coeflicient of : therefore in |. eae is —a, where a is a root of the equation ¢ (x) = 0.
Supposing the equation of ~ dimensions, which of its roots is given by this method? The result analytically comprehends all the roots, but arithmetically it gives only the least root. Let us recall, for example, the quadratic equation «°+aa2+b=0; and, supposing «, 8, to be the roots; —a=a+ , b=af; and substituting these values in the expression for the root given at the beginning of this Section, we get
aB | @B 4 dB 6.5 aif o+B (@+Bf. 2 @+py sear 3° (a+B) RQ
root = ——,, + &e.
132 Mr. Murpny on the
and expanding each term of this series according to the descend- ing powers of 8, the value of the root is
4 a 5.6 @& +3 = {1-5 B* 1.2 men Be 6.5 a‘ a 8.76 ,a tog 4° pu &e &e. &e.
All the terms here mutually strike out with the exception of the first a; but since a and # are similarly involved in the given series, it follows that if we expanded according to the descending powers of a, all the terms would strike out except 8; thus this series analytically represents a or ( indifferently; but if we stop
at the »' term in the former case, the error is of the order i) :
B and in the latter, of (f) ; and if a <8, the former is very small,
and the latter great; the series therefore, arithmetically, gives the least root (abstracting from its sign); and generally, whatever is the proposed equation, the series expressing the root is manifestly a symmetrical function of all the roots, and therefore does not analytically express one more than another, but comprehends alike all the roots; but as each term may be expanded in a con-
Resolution of Algebraical Equations. 133
verging form when we make use of the ascending powers of the least root, the given series arithmetically designates the least root. And thus it is that Lagrange’s series which, as it will be hereafter seen, coincides with that obtained by the present method, repre- sents arithmetically the least root of the proposed equation.*
SECTION II.
To find any function} /(«) of the root of a given equation p(w) =0.
From {(0) subtract the coefficient of - in Pen ae f(x) being the derived function or differential coefficient of f (x). the remainder will be the required function.
After the examples given in the former Section, many will not be necessary, in illustration of the present principle: we shall take but one, viz. to find the value of 2 when «=c..«',
n being positive. The required value of x", by the above principle, is the coeth-
s Te . cient of - Im — 227-11. (1 — <.'); and expanding the log. and ob-
* When the first power of x does not enter the given equation, put r=a-+z, so as to introduce a first power of z; and then, let z be treated as the unknown quantity. + f (2) is supposed to contain only positive and integer powers of x; should it be other- wise, we have only to put c=a +2; the same is to be observed of ¢ (a).
134 Mr. Murpuy on the
serving that the first —1 terms may be rejected as not con- taining any negative powers of z, we get
nm ett ct |
: Ls c a” =n x coeflicient of ; im }
nx n+1.x that is, nm.n+2 2.n + 3” m — an onthe ert? ort 4 &e, ar=c"+n 1 ete 1.2.3 Cone + We,
This principle evidently includes that given in the former section as a particular case, and admits of a proof nearly as simple, viz.
Put ¢(«) in the same form C.4—a.a-fP.x-y¥ &e.
= pia) . , ass = ah £ and .. l. pis LCS ee Oe: Fe ee
the part of which containing negative powers of < is
1 2 3 ee
xe
Hence = is evidently the coefticient of 5 im — vue ;
: us and .. a" = coefficient of pl ee mee 2),
Let f (ce) be any function of a, as f (0) + 4a+ Be’ + Ca’ && putting for 7 1, 2, 3 &c. successively, it follows that
J (a) =f (0) + coefficient of in —-{4+2B2r+3C2* &e.} sf. ei)
= f(0) — coefficient of F in f(x) .1. we
Resolution of Algebraical Equations. 135
SECTION III.
To find the sum of any specified number (m) of the roots of an equation.
Divide the given equation by x", take the Nap. log. of the quotient; the coefficient of 4 , with its sign changed, will be the required sum of m roots.
Ex. (1). 2° +ax+6=0, to find the sum of two roots, take the coefficient of Lint (1 Aap =) viz. — a, which is evidently the
x x x sum of the two roots.
Ex. (2). 2°—aa"—b=0, to find the sum of two roots. By the above rule, that sum
38 a = coefficient of - in — 1. ei — 5 x x
ax” + _)
ax’+b i. (ax" + by t, (a x” + by? x? oa 32°
Sad 1 = coeflicient of 7 + &e.
Hence, when x is even, this quantity is nothing which agrees with truth, since the roots of the proposed equation are then of
the form + ./a.
But when ~ is odd, the terms of this series which involve 1 +1 z are at m places distance from each other, and begin at the “>
term, its value therefore is
Ae 4 3n=1.80-3 @& a’ Wight 5n—1.5n—3.5n—5.5n—7 a’ pp"F ge
—$—$— $< $—$—$<—$—$<———————— ———__———"
Cw ay OM TAGS
136 Mr. Murpnuy on the
When the term involving 2” is wanting in the proposed equa- tion, then, instead of dividing immediately by 2”, put x= +a, and the difficulty of having no term free from the unknown quantity in the quotient (on which depends the expansion of the log.) will be avoided.
To prove the rule given in this Secticn, suppose a, a2..-an, @,4,++-a, to be the 2 roots of the equation ¢ (x)=0.
Hence ) (x)=C. (w—a,). (@—ay)......(@—a,,) (@ — ay 43) voeeee (w—a,)5
a ea (1-%)....(0-%)
x (1- z ys} ok (1-2),
An+1
and taking the log. of both sides, it is manifest that
(2)
ae
: es a,+a,+...4,,=coeflicient of ie if
This method then will give us very simply the sum of any proposed number (m) of the roots of the given equation ; but since there may be various combinations of m roots made from the 2 roots of the given equation, which combination or group of m roots is that given by the present rule? The answer is analytically, it gives any possible group of the m roots, but arithmetically the m least: that it analytically gives any is obvious, since the resulting expression is manifestly a symmetrical func- tion of all the roots, and therefore cannot analytically represent any one combination of m roots more than any other. That it gives the sum of the m least roots may be thus shewn.
Suppose, first, that the sum of two roots of the equation $(x)=0 is obtained by the present method, and that a,+ a, is the particular combination which it gives.
Resolution of Algebraical Equations. 137
p (2) oe
: ee Hence a,+«a,=coefticient of 2 Wise 1.
Now since a, is a root, therefore ¢ (x) is of the form x -a,.P;
e iL er: IZ therefore a, +«a,=coefticient of i ie = ae
P
: ie =a,— coefficient of - in I ae
but the coefticient of - in me is the least root of the equation
P=0 (by Section 1).
Hence a, must be the least of the quantities: a, a;, a,, &c.: by similar proof «, must be the least of a,, a,, a,, &e.; and therefore a, and a, are the two least of a,, a., a;, &c., that is, they are the two least roots.
In the.same manner if by the present method we get the sum of three roots a,+a,+a,, then putting ¢(«) =2—a,.%—a,.P’, we have 3
-1 8 =-1(1-*)-1(1-2)-12,
x and taking the coefticients of * at both sides by the theorems
already established, we have a, +a,+a,=a,+a,+least root of the equation P’=0; therefore a, is the least of the quantities a,, a,, a;, &e. “LOTTE, LEGA aaa un DS 1 MN eT 2, G4, as, We. Aiden audvesins chs as caetaesancedasnconvss @1, Ay, a5, &C.
therefore a,, a,, a,, are the 3 least roots of the equation: thus it appears in general that this method gives the sum of the m least roots.
Vol. IV. Part I. Ss
138 Mr. Murpnuy on the
In comparing thus the magnitude of the roots, we go on the
hypothesis of their reality. When the roots are imaginary how-
ever, a similar order may be supposed to subsist ; thus— «+ Wass ——b s a less numerical root than —a—‘\/ = — b (abstracting from the
sign) when 4 6, it holds therefore the same rank when +; <b.
Since the whole theery of imaginary quantities results from an extension of the properties of real quantities, we have this advantage resulting from the theorem above proved, that it esta- blishes an order when quantities are some real and some ima- ginary—analogous to He relation of greater and less amongst real quantities.
To find the m™ least root of a proposed equation ¢ (2) = 0.
From the coefficient of = - inl. ote) subtract the coefficient of } ~
am 1 = )
in 1. , the remainder will be the m"™ least root. This theorem
is an esac consequence of what has been already proved in this Section.
To find the sum of any function /(a,) /(a.) &. of the m least roots.
From m/(0) subtract the coefficient of * = im of (a). pec )
a ? the remainder will be the required Fee
The proof of this theorem is similar to that given for a function of one root, it will be unnecessary therefore to add it.
To find the value of any function of the m least root.
Resolution of Algebraical Equations. 139
Find, by the last theorem, the sum of the values of that func- tion for the m least roots, and also for the (m—1) least roots, and take the difference.
SECTION IV.
In this Section we shall make a few applications of the present method, to shew with what facility it gives various theorems of Analysis.
Lagrange’s Theorem.
Let s=a+h F(z) to find f(s) any function of x; put s=2+ a4, the equation becomes x= F(a +2); to find /(«+ a), apply the rule in Section (2).
Hence °
hFa+x fe
F(z) or f(a +a) =f (a) — coefficient of + in f'(a+2) ie
=f (a) + coeflicient of in
h 2 fi ad be See oe ; 7S (a+ 0). Fase) Ces. eee We.
|
5
Now if we consider f(a+x) F(a+2), f'(a+«)F(a+2), &c. as so many separate functions of @+., and expand them by Taylor’s Theorem, it is visible that the above equation becomes
S (x)= fla) +h f' (a). F(a) + i ee poe &e.
s2
140 Mr. Murpny on the
Laplace's Theorem.
Let x= F§a+ho(z)} to find f(2).
Put s=F(a+u) the equation becomes w=hd.F'(a+u), to find SF (a+u), or f(z) apply the rule in Section (2).
Hence
f(s) =fF (a) — coefficient of + in fF(a+u)l. 14h AES
Uu
and expanding the log. as in the last case, it evidently gives
SO) = fF (a) + hf F(a).p Fla) + 25 (fF la). oF ay + &e.
Burmann’s Theorem consists in this, that if x and uw are two functions of « which vanish together, and X any other function,
then f. ; ee de (a) 4 du" asa
putting x and w = 0 after the differentiations.
ts In fact, if we form the equation «= X"*', the value of a is
’
1 coeflicient of : in -1(1 es -)
therefore coefficient of h’+! in u
' ly: A dX i = coefficient of - m (+1) .a* = i ae ei when wu is put = 0 by Maclaurin. Put the same equation under the form Bhs s=h.-.X¥™, Uu then by Section (2) 1 axe hi es
phe y u or AX" = coefficient of 4 in - 2 L(l1--.- 2X"); 3 dx BU
Resolution of Algebraical Equations. i41
*. coefficient of A"*' in u
n 1 er Be ea pa ann = coefficient of = in — . : é (-) % gr n dx u pn - 1 OX ax” = coeflicient of z"-' in ——— ,-—< , (¢) m.n+1 4 u
nisdX (z%\") moe, 1.2...n+1dx"-*"
Equate this with the former value and we get the above theorem.
To find the sum of any function of the m least roots of the
equation (3 — a)" =h. F(z) as f(a,) aia (cre) etna ST (an): Put S=x+a; -. a@=hF (a + 2) to find
J (a + By) + f(a + Br) + &e. 1, Bx, &e. being the m least values of x, we have, by Section 3, the required sum
= m f(a) — coefticient of 4 in f(a +2) 1§1— L F(a+x)t:
whence, if we expand the log. this becomes
ae f(a). Fiat ht de ae (a). F(a) } = h. : a BT (a) +h 1.2...m—1.da™-} at 2°1.9...9m—1.da"— a
In this remarkable theorem if we put m=1 it becomes Lagrange’s; but if we put m= the dimensions of the equation, then the m least are in reality all the roots, so that this case corresponds to the theorem given by M. Cauchy in Vol. 1x. Memoirs of. the Institute.
142 Mr. Murpny on the Let us retake for a moment the same equation (zs — a)” = h F'(z),
if we extract the m™ root of both sides and suppose 1, p, p’---p”~'
to be the m" roots of unity, we may form m different equations, viz.
1 St + ped” ae (e)*
to each of these equations apply Lagrange’s Theorem to determine f(z), we thus get m values, viz. :
S@)=f (0) +0. f a). Fay +5 Af (@) Fary'+ be.
SF (s) =f (@) + phng’ (a). F(a" = eg {f (a) Fy +&e.
1 1 o.m—1 a 30 Ae SF (8) =f (a) + p"-. hf (a). F(a)” + = if’ (a) F(a)"}' + &e. Adding up all these equations, and recollecting by the proper- ties of the roots of unity that their sum, or the sum of any of their powers is 0, except in the case of the m", 2m", &c. powers.
when the sum is m, we evidently get for the sum of the function
Resolution of Algebraical Equations. 143
of the m roots precisely the same result as before obtained in p. 141.
We see likewise from this, that if there is an equation of x dimensions, and we transpose the highest power, so as to put the equation under the form «x = %/P, then applying Lagrange’s Theorem, to find x, and multiplying the first group of x terms by
Qn ear, A — .4r , 1, co Nem cos = + /—Isin—, Xe.
and the 2”, 3", &c. groups of x terms respectively by the same, we shall get a 2" root of the equation, and multiplying them by
4a — . Ar 118 cost nf =I a ae &e.
we shall get a 3" root, and so on for all the roots.
Suppose a series is to be reverted, or an equation solved, such as a,=«—a,2°—a,«°, &., we can easily express the law of the general term in the value of «, thus by Section I.
ae | Pe a A «x = coeflicient of = in—l. Sy — (% + '4,0+ 4,2", we.) x ? x 4 A 3 ey lie therefore the general term of «=coeflicient of > in X (* + A,X 4+ A,X", &e.) 3 NNV2X
(“ +4,%+a,2°, &e.\" ‘Vy =coeflicient of a” in € * 67%. 67%", Xe.
Now
1.2...”
Hence the general term of «=coeflicient of 1.2...x—1 i in the fol-
lowing product, viz.
2
ad,
aa,
is
(1 + ast + — &e.) (1+aa,0° + &e.),
144 Mr. Murpny on the
, poy LS mal. aya &e. 1. €. W=27 ob, x 1.20 b es
subject to the two conditions b,+6,+ 6, &e=n, b,—6,-—2b., &e.=1. When to an equation F'(x)=0 we add a very small term e.¢9(), what is the corresponding correction to be applied to the root? The root of F(x) =0,
is the eoeticient of : in -— 1. erie)
and the root of F'(x)+e¢(x)=0 is the coefticient of 2 ened (F(a) + Cpr) ; 3 x l « x § therefore the increment of the root
=edamaentiof ano (= “s a) cay Fa x xv x
ep (x) ) F(z) \
= coeflicient of s in 1. 1 +
eg («) F (x)
= coefficient of 5 in when e¢ (x) is supposed to be very small.
Cor. When the proposed equation is s=a+/(*), which by adding an additional smal] term, becomes
x=at f(s) +eh (®)
the correction to be applied to the root is
e ih (a) + [p (a) fa] + (oa) fer’ + &e.{.
Resolution of Algebraical Equations. 145
To find the value of the error committed when we stop at the x“ term of Lagrange’s series.
If we pursue the same method as that used at the beginning of this Section, we get the required error
hn?
= coefficient of = in J (a+2).F(a+a)*?
m+1.art}
Arre Sareea. (a+2).Fa+x"*", &e.
1
a : i \ (4. F(a+2)\"*? = coeflicient of ~ in f (a+2) (=) ra
piesa ae of
Lv “n+2
sate 1. : FE (a--a)\"*} - =coeflicient of Z in f (ata). f )(—<**) ch
+ (Fy et bat
the integral commencing from h=0.
But the part under the sign of integration may be summed, and (eae ig hn — o ‘
= a ACES) x
_ F'(a+a)"*! h h 7, a ‘e@—h F(a+2)’
therefore the error = coefficient of «”~" in
he f (ata). Faray (ae,
nh
to be calculated by Definite Integrals, in the next Section. Vol. IV. Part I. Ab
146 Mr. Murpruy on the
SECTION V.
Ler F(x) be any function of x, containing only integer powers of 2, positive, or negative; as
F(«)=A + Br + Ca* + &e.
b ape ae & + &e. x x Put < for x, multiply then by ¢’ and integrate with respect to 9, we thus obtain
Be? Ce 9 ct ae + &c.
[ F'(e). = const. + Ae? + “6 : +b.0—ce-°— &c.
All the terms in this result, except 6. and the const., are cir- culating or periodical terms; i.e. if we give @ the series of ima- ginary values comprised in the formula @+«a,/—1, then for any two values of a, which differ by (27) a whole circumference, these terms have precisely the same value, but the term 5.0 be- comes 27b,/—1 between the above limits: thus it appears that the coefficient of the first negative power of x in
F(z) = Ss MF). €:
the integral being taken through one entire circulation, 1.e. from 9 to 0+ 27f—1.
Restore now for ¢ its value x; then /,F'(e°).e? becomes /, F(x) which by actual integration is
Resolution of Algebraical Equations. 147 fconst. + 4.2 + B.= + &e.
+b.1. (x) — “ — &e.}.
And the limiting values of «2, which correspond to the former limits of @ are e’, and &*”?*=?, both of which are equal. But the integral does not vanish, though the numerical values of the limits are equal; for it has been above shewn to be equal to
2xb./—1, which may be easily explained ; thus, Ax or Ae’ between limits= 4e'*?*”-*— Ae’ = Ae’. (e7"-1_-1) =0.
eat! Br alt Similarly —* between limits=0, &c.
but 61. 2 between limits=d 1. &*?*”~'_—51., é
=b (04+22,/—1)—be =2rb./—,
though, therefore, the quantity under the transcendent sign of log. is