119 



Employing therefore for — the development 



which is of the homogeneous form required, and, after differentiating 

 for 7, making «* + /3* + y* = 1 , we find for the ordinate Z^ 



Z = - ^ + J (1 _ y) ~ y 2,^ ^ (2/ + 4) w^.^^{a^ + ^=)H- > , (S") 



a series of which the term — A being the ordinate of the central 

 focus, the remainder is the longitudinal aberration : y is the cosine 

 of the angle which the near ray makes with the central ray, and 

 a* + /3* is the square of the sine of that angle. If therefore we de- 

 note the aberration 2^ -j- ^ by A, we may develope A in a series of 

 the form 



A = Z («H ^3=) + L, («' + ,3^)^ + &c. ^T") 



in which 



/,=-. i.4— 4w^, I,, =1^4 +2tt>, — 6«;„. ' (U") 



And if by these relations (C/"), we eliminate w^ xi\ from the ap- 

 proximate expression (E"), we fmd the following formula : 



IJ-w' C 2L, —d L 6L°- » 



"*" 12{« + y})5 I z + A '^(z+Af'^*\' (^") 



which shews the connexion in a system of revolution between the 

 development of the longitudinal aberration A, and that of the cha- 

 racteristic function V. 



