1*7 







IH 



If. any of the preceding, the tonn involving I be K-ft out, we hare 

 Ik* eoouMun innriiilsMli mode of determining the focus. We now 



i to the lonnubs for determining the aberration in the cue of a 

 direct pencil of parallel rays. 



Lit M .. in the thud fifure (the perpendicular distance from x to the 

 uia will do equally well; be -y; then AF U determined by the 



-oj 1 a + 





/I 



(i 



l U' 



+ 7) 



where B U the radius of the surface which the light first meet*. The 

 algebraical value of MA in the figure i* r + K, where r is supposed 

 corrected for the thiotnae* If wa aatume 



when A stand* for 





The following are the results, assuming p = z, which is near enough 



(if the material be glass) for determining this correction. This 

 supposition give* (x being the numerical value of x, independently of 

 aign) 



7x'-10x + 10 y 



K = - . -. 



(1). Plano-mitnx : x= -1 K = jj-' 



(1). 



: X=l. 







x 

 (3). Coxrtxo-eoncarc t*txuctu ; 



: light entering at the more convex side : 

 i-r 7 a? 10 J- + 10 j/ 1 



: light entering at the less convex aide : 

 i 7 x 1 + 10 ,r + 10 y* 



r 10 



6 



(8). Coneoro-roiirtc ateaaetu : 



* m :r-ti K = - 



c-rlO 



(4). PloM-aM>r : came * a* plano^onrex. 



(4). Omcaro-piane , ume at ooorexo-pUoe. 



(5). Amok euKtarc, light entering at (he more concave aide : lame 

 *i double OOOTCX, light entering at the more convex aide. 



(5). DoukU eoneave, light entering at the let* concave aide : aune a* 

 doable ooOTex, light entering at the lee* convex aide. 



(). 0mmzMam . amme a* concavo-convex meniscus. 



(4). C<mfaro*fmrtx : lame a* convexo-concave meniacus. 



Throughout theae formula) the aign of K ia oppoaite to that of F. 

 ataee 1x> 10* + 10 mart alwajn be poiitive. Hence the p-.int A 

 alwavi lie* between the point* X and r (third fig<ire). If the iK.int x 

 be placed a* high ai poanble, that i, if y be what ia called the irmi- 

 aperlm of the lena, then K ia the aberration of the extreme ray. It 

 afpean al*o that the longitudinal aberration varies directly aa the 

 quare of the nemi aperture, and invenely a* the focal distance. 



The aberration U least for a given aperture and focal length, when 







* 7 "hick give* x 6 B, requiring a double-convex or double con- 

 cave lens, in which the radiii* of the aide on which light enter* ia one- 

 sixth of the other. The convex lew of thi* kind U what optician* call 



the crowed lens. The co-efficient of f+i is - 1 



The latitudinal aberration at the focus (a. determined with the 

 correction for the Uuekoes.) is y K+r, or (neglecting the aign) 



g A {3; forglas. - 





|B . * T" d fi r re ( ' nd thc """o m y 



? sun light thrown into an otherwise dark 

 through a oonvn lens) we ahull see that thc luminous (pace U 

 bounded by a surfso* of revolution which narrow, and afterwardi 



spreads again, a* in thia diagram. The smallest circle (at o) i* called 

 the circle of least aberration, and U determined as follows :- -Its 



centre U nearer to thc glass than the focus (corrected for the thick- 

 si) by three fourth* of the longitudinal aberration of < : 

 ray; and its diameter is one-half of the lateral aberration of tlie 

 extreme ray. If then we measure from the corrected focus, we find 

 for the distance of the circle of least aberration (neglecting its sign) 

 from this focus, 



3 Ay' 7x*-10 : 



"327 ; for 8'"" 



and for the diameter of thia circle, 



197 



for 



7x J -10x-rlOy 

 -- 12 -- P- 



The correction for the thickness, to be subtracted from T aa deter- 

 mined by the first equation of all, is 



- ^ (x + !).<; for glass- ^(x-rl).<; 



which is always algebraically subtractive, whatever the sign of P may 

 be. The following table exhibits thia correction, the distance of the 

 circle of least aberration, and its diameter, for the cases above noted. 

 The description of the lena is in the first column, and I stands for 

 plane (or piano), c for concave, and inverted c for convex. The aign 

 of the aurl'ace which the light first meets is placed first. Where a 

 great and small letter meet, the small letter shows the aide which has 

 the less curvature, or the larger radius. 



o = 7 3? lOx + 10 

 = 7 a? + 10o: + 10 



Wo have judged it more useful to collect what we may call thc 

 rrilicnl formulie, by which the fitness of a lens for any given purpose 

 may bo estimated, than to enter upon explanations of optical prin 

 in an isolated article. We shall now give the formula) only, omitting 

 ill of cases, when thc pencil of raya is not parallel, but proceeds 

 from a point in thc axis. 



Let c be the distance of the focus of the entering pencil from the 

 surface whose radius is n, and v the distance of the focus of ti 



thcr side from the Mirface whose radius is a ; U beii 

 wl'ii t t, and v 11 



-:ent. Let r be the distance < 



illel rays from thc surface of emerges nod as 



Then, if the thickness of the lens be inconsiderable, v ia deter- 

 mined from u by the equation, 



- - - - /M . !\ 



v representing the solution of thia equation, the more correct value 

 taking the thickness I into account, is 



sn-\ _ iy^v_ 



" \ ' f ' 



Let! 



T p 

 + n ; w= v + u' 



then the above correction for thc thickness ia 



c-w 



