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61. Silene inflata, Sm., D. C., &c.

Var. intricata nob.; vix glaucescens; caulibus ramosissimis, dense intricatis, elongatis, pendulis, basi suffrutescentibus.

Hab. in rupibus excelsis declivibus Convallium Maderæ.

FRANKENIACEÆ.

62. Frankenia cespitosa, nob.

F. caulibus fruticulosis, ramosissimis, densissime cespitosis, humilibus, humifusis, calycibusque basi velutino-pubescentibus: foliis sessilibus, basi connatis, linearibus, glabris, margine revolutis, basi breviter et parce ciliolatis floribus in capitulos terminales congestis, subcymosis, foliis multo longioribus.

Hab. in collibus maritimis sterilibus aridisque Promontorii Ponta S. Lourenço Maderæ; etium Portûs Sti.

F. ericifolia C. Sm., necnon F. corymbosa Desf. nimis forsan affinis.

63. Viola paradoxa, nob.

VIOLARIEÆ.

V. suffruticosa, e basi ramosa; ramis subproductis, elongatis, simpliciusculis; inferne nudis, stipulisque simplicibus linearibus integris minutis sæpe obsoletis, petiolisque elongatis marginatis ternato-fasciculatis, foliisque rotundato-spathulatis crenatis basi abruptis cordatisve, glaberrimis: foliis summis cuneato-elongatis, in petiolum attenuatis, apice subtridentatis, petiolisque caulibusque subpuberulis: sepalis oblongis integris, bracteisque pedicelloque pubescentibus: calcare obtuso, calyce longiore; nectario...... capsula obsolete hexagona, glabra; seminibus pallide flavescentibus, paucis (15—20), ovatis.

Hab. rariss. in summis cacuminibus montium excelsiorum Maderæ, in fissuris rupium. Invenit cl. Car Lemann, M.D.

Obs. Cum V. calcarata L. conferenda. Flos aureo-flavus. Stylus ab apice ad basin attenuatus. Stigma urceolatum, utrinque fasciculatopilosum, ore magno, expanso dilatato, inferne in labellum producto. Folia ad apices ramorum sterilium conferta. Pedunculi solitarii, axillares, subpollicares. Capsulæ abbreviatæ, obtusæ. Semina, præter colorem, fere ut in V. tricolore L.

CRUCIFERÆ.

64. Sinapidendron salicifolium, Prim. p. 37. No. 65.

Syn. Sinapis angustifolia, D. C. Prodr. 1. 220.

Hab. in rupe quadam excelsa maritima, "Cabo Giram" dicta, prope vicum Camera de lobos Maderæ, nuperrime ab amico Rev°. M. Tucker, botanophilo vel oculatissimo, detecta. Species genuina videtur.

65. Matthiola maderensis, nob.

M. biennis: caule herbaceo, erecto, elato, ramoso: foliis oblongis, integerrimis, incano-tomentosis; radicalibus densissime rosaceo-confertis : siliquis compressis, glanduloso-muricatis.

Hab. in rupibus maritimis Maderæ et Portus Sti. ubique vulg.

Flores pallide violacei, vespere præsertim odori, rarissime albi. Species intermedia, habitu foliisque M. incana R. Br.; siliquis, 3-5 poll. longis, M. sinuatæ, ejusd.

SUPPLEMENTUM.

MADERA, October 1837.

66. Chara atrovirens, nob.

CHARACEÆ.

C. atroviridis, pellucida, lucida, gracilis, fœtens: caule ramisque tenacibus, tenuibus, flexilibus, hinc inde articulatis superne sparsim minutissime retrorsum papilloso-spinellosis, contorto-striatis: ramulis verticillatis, 5—6-articulatis; articulis inferioribus striatis; summis simplicibus, bracteisque 2-4 inarticulatis, cylindrico-setaceis, nucula triplo quadruplove longioribus, lævibus, haud striatis, glabris.

Hab. in rivulis aquarum fluentium Madera rariss.

C. vulgari Ag. proxima. Differt colore, scabritieque ramorum, radicellos nascentes deflexos, sparsos æmulante.

MADERA, May 1838.

XXVIII. On a New Correction in the Construction of the Double Achromatic Object-glass. By RICHARD POTTER, ESQ. B.A. Queens' College.

[Read April 30, 1838.]

THE achromatism of the compound object-glass of telescopes has never, that I am aware of, been investigated otherwise than for very small pencils.

Sir John Herschel, in his elaborate and excellent paper on the aberrations of compound lenses and object-glasses, published in the Transactions of the Royal Society for 1821, when noticing the investigations of Clairaut, Euler, and D'Alembert, makes no mention of any higher approximation having been attempted; and he himself follows no other method for the chromatic dispersion, although he has pursued the subject of the spherical aberration so far as to render the objectglass free from it, for astronomical and terrestrial objects at the same time. Indeed, from the following passage in the same paper, it is clear that he did not suspect the existence of any unconsidered residual dispersion, of the magnitude of that which I am about to discuss; for he says, "The simplest considerations, indeed, suffice for the correction of that part of the aberration which arises from the different refrangibility of the differently coloured rays; and accordingly this part of the mathematical theory of refracting telescopes was soon brought to perfection, and has received no important accession since the original invention of the achromatic object-glass."

It must have struck most persons conversant with the subject, that the effect of the lenses, in an achromatic combination of two or three lenses in contact, must be sensibly different near their edges, on account of the oblique passage of the rays, from what it is near their centers; and this difference will be the more important, as the area of that part of the surface of the lens, with this unconsidered effect, is so much more than that part of the surface near the center for which the common theorem is accurate, or nearly so.

In the present paper, I have investigated the conditions of achromatism in a double object-glass for a ray passing through it at a distance from the center of its aperture, on the supposition that we may neglect powers, of the small quantities which enter the expressions, above the first, and also their products. It is also necessary to consider the thicknesses of the lenses, as that of their edges, for all parts at which the new correction rises to any important magnitude.

I have arrived, by two different methods, at the same result, which involves the expression obtained by the ordinary mode, together with others depending on the thicknesses of the lenses. The spaces, through which a ray has passed within the lenses, have on the achromatism an effect which is precisely similar to that of the distance of the lenses in achromatic eye-pieces. If the lens have great thickness, a ray of light after an oblique passage through the glass, will meet the second surface at a different angle to what it would have done if that thickness had been small; and hence, if we consider a virtual prism to be formed by the tangent planes to the surfaces of the lens, at the points at which the ray is incident and emergent, the angle of this virtual prism will depend on the thickness of the lens, as well as on the radii of the surfaces and the distance of the point of incidence from the center of its aperture. We may easily conceive, that this variable angle of the virtual prism will need more accurate consideration when we pass beyond the ordinary first approximation.

The new correction, which we thus arrive at, supposing the thicknesses of the lenses in a double object-glass such as might arise in practice, is however not very large in magnitude. But nevertheless, if

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