of Refraction at one or more Spherical Surfaces. 487 



from its conjugate P', the rule of signs being that already 

 referred to (§ 1). 



5. Fig. 2 exhibits the construction adapted to formula (2). 

 P in one axis coincides with its conjugate P / in the other, and 

 the line joining any other two conjugate points Q, Q' on the 

 two axes passes through the point (d, df). 



If the origin be the self-conjugate point 0, the centre of 

 the sphere, the relation (2) becomes 



p p 



where (fig. 3) 



OF=/ / , OP=p, &c. 



As in § 3, we have 



6. In the following proposition, which, in the form given, 

 is due to Helmholtz (vide Optique Physiologique, p. 72), I have 

 changed his notation and employed the rule of signs (§ 1), in 

 order to exhibit the result of the elimination in a symmetrical 

 form. 



Let there be any number of spherical refracting surfaces 

 whose principal foci are (F l3 F^), (F 2 , F^), &c, and which 

 cut the common principal axis in A, B, C, . . . Let (R , Hi), 

 (Hi, R 2 ) • • • be pairs of conjugate points such that EoF^^, 

 RiF'i^x, ... In like manner let (P , P x ), (P x , P 2 ) . . . be 

 any other set of conjugate points, such that R P = po> 

 BaP, =p' 1) ... Then, by §4, 



p p\ 



^+^=1,&C. 



Pi Fa 



Also by the rule of signs (§ 1) we have^x= — p\, p 2 = —p' 2 , • • • 

 Hence, on eliminating these quantities, the position of Y n , the 

 point conjugate to P with reference to the system, is deter- 

 mined from an equation of the form 



£ + ^-=1, (3) 



PO Pn 



where /= R F, / = R n F'; F, F' being the principal foci of the 

 system. /• 



The values of ^- for 2, 3, 4, . . . refractions are, respectively, 



di did 2 d x d 2 d 3 



h-hd'i d x d 2 -\- d \d 2 + d \d' 2 d x d 2 d z + d\d 2 d d + d\d' 2 d % + d\d r 2 d' z 



