292 • Sir W. R. Hamilton on the Focal Le?igths and 



st( 4 ) 



-V( 1+ ^) = + * + ^ ; ••• < 18 -' 



and therefore, taking the reciprocals, 



1 a' / a' 2 \ a' ST (4) , ■. 



-z = t( 1+ it) + i^ ; •••• (19 ° 



Adding these last two equations, attending to the value (16.) 



of £, and observing that T^ ', after substitution of that va- 

 lue, becomes a homogeneous function of the fourth dimen- 

 sion of a' and (r, so that 



^> + ^) =4tW) (2L) 



! r -(,- 1) ,-=i(«-3-^) + *T(«. m 



la! 8 <r 



we find this relation : 



f_ 1 



And changing, in the second member, a! and er to their ap- 

 proximate values given by (19.) and (20".), namely, 



cl m ^ X <r = - ^, (23.) 



we find 



/x z 



; 2 1 



.(24-.) 



By suppressing the second member, we get the intersection 

 of the axis with a refracted ray infinitely near it, or the ex- 

 traordinary focus of the central rays; by taking account of 

 that member we get the longitudinal aberration. 



6. As a verification, we may consider the case of an ordi- 

 nary refraction at a spheric surface as being included in the 

 foregoing, and the formula for that case must result from the 

 equation (24.), by making therein 



v = ft, S- — r 3 (25.) 



