CHAP. 8. GENERAL PROPERTIES OF FOCUSES AND IMAGES.

CHAPTER VIII.

149

Contains general properties of focuses and images, belonging to the eye and to any number of mediums: with general conftructions, fhewing the variations of the apparent diftance of an object, and of the real distance of its laft image, from the eye, caufed by a direct motion of the eye, object or mediums.

H

PROPOSITION I.

AVING the diameters and pofitions of two spherical furfaces, which Fig. 383. intercede three given mediums, and fuppofing the incident rays in either of the outward mediums to be parallel, and very near to the common axis of the furfaces, it is propofed to find their focus after both refractions.

366. In the common axis AC of the furfaces AB, CD, let a and d be the focuses of rays, which before they emerged by refractions at AB and CD into the outward mediums, went both ways parallel to the axis in the inward medium. Moreover let b and c be the focuses of other rays, which before they emerged by refractions at AB and CD into the inner medium, came both ways parallel to the axis in the outward mediums. Thefe focuses may be found by art. 224. Then fay as cb:bA:: Aa: aI; and placing a I the contrary way from a to that of be from b, the point I fhall be the focus of the rays after both refractions, which came parallel from without upon the furface CD: becaufe c is their focus after their firft a Art. 237. refraction. In like manner fay as bc: cC:: Cd: dK, and placing dK the contrary way from d to that of cb from c, the point K fhall be the focus of other rays after both refractions, which came parallel from without upon the furface AB.

All the figures are adapted to mediums whofe denfities are continually greater as they lye in order from the left hand to the right; but the demonftrations serve for any irregular order of densities.

PROPOSITION II

The focus of incident rays being given, it is proposed to find their focus after their refractions at two spherical furfaces, which intercede given mediums.

367. In the common axis AC of the given furfaces AB, CD, leta and Fig. 384d be the focuses of rays, which before they emerged by refractions into the outward mediums, went both ways parallel to the axis in the inner medium. Moreover let I and K be the focuses of other rays, which be

fore

Fig. 385.

a Art. 237.

Fig. 386.

fore their refractions through both the furfaces, came both ways parallel to the axis in the outward mediums. Then let P be the given focus of the incident rays, I and a the focuses of thofe rays which came the contrary way to the incident rays, and fay as PI to Ia fo dK to KR; and placing KR the contrary way from K to that of IP from I, the point R fhall be their focus after both the refractions.

For by faying as Pa:aA:: Ab:b2 and by placing b 2 the contrary way from 6 to that of a P from a, the point 2 is their focus after the first refraction at AB. The fame point being the focus of incident rays upon the furface CD, fay again as Qc:cC:: Cd: dR; and placing dR the contrary way from d to that of c 2 from c, the point R is their focus after both refractions. Now by the first of these proportions, and of those in the foregoing propofition, the rectangle Paxb2= (a A× Ab =) bc ̧x Ia; and by the fecond proportions in this and in the foregoing propofition, the rectangle 2cxdR= (cCx Cd=) bcxdK; then by refolving the two former rectangles into the proportion of their fides, it is as Ia: Pa::b2be and disjointly (or conjointly) PI: Pa:: (Qc:bc::) dK: dRby refolving the two latter rectangles; and disjointly (or conjointly) PIIa:dK: KR.

368. Corol. The magnitude of the rectangle under PI, KR is inva riable, being always equal to the given rectangle under Ia, dK; and confequently KR is reciprocally as PI

PROPOSITION III.

Having the diameters and pofitions of three spherical furfaces, which intercede four given mediums, if the incident rays in either of the outward mediums be parallel, and very near to the common axis of the furfaces, it is propofed to find their focus after all the refractions.

369. Suppofing the focuses a, d and I, K to be determined by the foregoing propofition for two contiguous furfaces, as AB and CD; let parallel rays falling on both fides of the third furface EF have their focuses, after refraction there only, at e and f; whereof e being the focus of incident rays upon the furface CD, fay as eK: Kd::a I: IL; and placing IL the contrary way from I to that of Ke from K, by the foregoing propofition the point L is the focus of rays, after all the refractions, which came parallel from without upon the furface EF. Again, fay as Ke: e E :: Ef:fM, and placing fM the contrary way from f to that of eK from e, the point M is the focus of rays after all the refractions which came pa› Art. 237. rallel from without upon the furface AB..

PROPO

PROPOSITION IV.

The focus of incident rays being given, it is propofed to find their focusafter refractions through any given number of spherical jurfaces that intercede given mediums.

370. Cafe 1. In the common axis of three furfaces AB, CD, EF, let I Fig. 3876and fbe the focuses of rays, which before they emerged by refractions into the outward mediums, went both ways parallel to the axis in either of the inner mediums, as CE; moreover let L and M be the focuses of other rays, which before their refractions through all the furfaces, came both ways parallel to the axis in the outward mediums; then let P be the given focus of the incident rays, Land. I the focuses of thofe rays which. came the contrary way to the incident. rays, and fay as PL: LI::fM: MS; and placing MS the contrary way from M to that of LP from L,. the point S fhall be their focus after three refractions.

For by the fecond propofition their focus R, after two refractions at Fig. 388. the furfaces AB, CD, is given by faying as PI: Ia:: dK: KR; and fince R is their focus when incident on the furface EF, fay as Re: eE:: Ef: fS, and S is their focus after three refractions. Now by the first of these a Art. 237, proportions and of those in the third propofition, the rectangle PIXKR = (IaxdK=) Kex LI. Likewife by the second of thefe proportions and of those in the third propofition, the rectangle RexƒS=(eExEf=) Ke xfM; and by refolving the two former rectangles into the proportion of their fides, it is as LI: PI::KR: Ke, and disjointly (or conjointly) PL PI::(Re: Ke:)ƒM:ƒS by resolving the latter rectangles; and difjointly (or conjointly) PL: LI::fM: MS.

The rays whofe focuses are I and ƒ were fuppofed to be parallel to the axis in the medium CE; now let a and t be the focuses of other rays which are parallel to the axis in the other inner medium AC; and fince PL:LI::ƒM: MS; for the fame reason when P comes to a and confequently S to f, it is as a L:LI::ƒM: Mt, therefore because of the given rectangle LIxfM, it is as PL: La:: tM: MS.

371. Cafe 2. Hence by the method of the third propofition one may find the focus of parallel rays after refraction through four furfaces, and then the focus of inclined rays in the fame manner as in the foregoing cafe, and so on. And by these propofitions it is fufficiently evident, that if L and M be the principal focufes of the whole fyftem of furfaces, and I and fbe the focufes of other rays that go parallel in any one of the inner mediums, then PL:LI::ƒM: MS.

Fig. 389,390

372. Corol. I. It is manifeft by the analogy between the rules for finding the focuses of rays refracted through a fingle furface and a fingle lensh, b Art. 236, that the rule in this propofition for the focufes of rays refracted through any number of furfaces, will ferve alfo for any number oftenfes of any fort

placed