i-ai 



which is to be satisfied for all the points of the refracting surface, and 

 which may therefore be differentiated, considering A F as a function 

 of z, ti, namely the difference of the developments (V") (X"), and 

 considering z as itself a function of »?, assigned by the equation of the 

 refracting surface (ff"). In this manner we find, transposing the 

 symbols A, d, 



dti dn 



d_F 



dz 



. ^d'V, „dz d^V , (dz\ 



■dz\^ d^V d>z dV 

 ^l? + -d7 ^ 



dz ' 



0- a'^^J.'^'^^ a '^'^ 



dz\^ 



+'(£) 



d'V /dzY ^ 

 dz^d„ + W J ^ dz^ 



(A'") 



^ d^,..dz_d^ dyV dh dV 

 "^ dn' ^ dzdn "^ dft dr? ^ rfa» + dn^ dz 



and, making after the differentiations >? == 0, we have 



_ dz d'z _„ d^z 



"-'''" TT: — "'' TaIS '^^s 5 -rT = bz^; 



d^ 



dn' 



dr? 



dV d'V „ d^V 



d;^ 



d!^ 



dV 



^^=^^' z+A 





F- 



dzdn 



d'V ^ |U,Z. ft 



'^ "^ ~ '^ 12(2 + AY ~ 4(z+^f 



1- 



(z + ^f (/s'ttfi (z + Af J 



; V (B'") 



d^v 



d^V 



dn^ (a 



+ Af \ 2(z 



2L, — 5L 





l(z + 



SZ," 



+ ^) ^ (3 + AY 



H-.). 



We have therefore, I", for the change of — A, the ordinate of the 

 central focus. 



VOL. XVI. 



