On the Estimation of Aperture. By the late G. HoeMn, jun. 339 



of emergent rays ; P Q and p q being indefinitely small (Plate VII. 

 fig. 1). 



Let P S^ s, Q S g' s represent the course of two rays through the 

 instrument, and let n, n! be the refractive indices of the media in 

 which P Q and p q respectively lie. 



Then because every ray from P passes through p we must 

 have 



w.PS — Sas — re'jis = a constant for every ray through P ; 



also 



w . Q S — S 6s — n'g s = „ „ ,, Q. 



Here S a .9 and S & s represent the " reduced path " of the rays 

 between S and s, or the sum of the product of the lengths traversed 

 in each medium they pass through by the refractive index of each 

 medium. 



Again S a s = S 6 s because these two rays start from and end 

 at the same point and make an indefinitely small angle with each 

 other. 



. • . M (P S — Q S) — m' (j9 s — ^ s) = constant, 



or, in the limit, if u, v are the angles that S P and s p make with 

 the axis, 



w . P Q sin u — n' p q sin v = constant 



= because u and v vanish together, 



x\gain, if N is the magnifying power of the instrumeDt, 



i9^ = N.PQ 



and n sin m — w' N sin w = 



sinw _ re'N 

 sin V n 



Had we assumed that rays from P converged to ^, and that rays 

 from P' converged to p' where P' and j?' are points near P and ^, 

 but situate in the axis, we should have found 



'^2 /W-n' 



where M is the ratio of the distances of two points measured in the 



direction of the axis to the distances of their images. 



It is clear that the two laws A and B cannot hold at the same 



n 

 time unless Vj = v, and then N = M = — and the instrument has 



n 



2 A 2 



