Aberrations of a Symmetrical Optical Instrument. 217 



We shall now consider the union of rays in the primary 

 and secondary planes respectively. 



►Since in equation (16) X/^ 1 = /i/y 1 we must have in primary 

 planes fi/X = ?7i//z and for secondary planes % 1 \ + r) l fjL = 0. 

 Let p Asi denote the aberration from the image-plane of the 

 primary focus of a small pencil proceeding from a point 

 distant # t - from the axis of the instrument and Si from the 

 ith. surface, and whose chief ray is incident on the latter at 

 a distance H t from the axis (where now H i 2 = ^ i 2 -\-r} i 2 ) . 



Then, taking for convenience the primary plane as the 

 plane of xz, the equation of the refracted ray being 



h ~ 2 n X-H x - 



H, 2 ^'-roV-IV 



we must have for primary foci 



(,/+AV- ]£)(*,' + 8.*/ -HO 



— fT2 + ^ 



stationary for small changes in H t . 

 This expression simplifies to 



V tf—n Si $i / l \ s^ s^ I 



Differentiating, we get 



p A, t - -A Si - ^j + H.J jg- - (qt + H,^, 



where \'=\/h%. 



Taking y t - and rj i zero in equation (9) and writing H, for 

 ? i? (9) may be written 



Hence we easily find that 



^^'_^^ = _s , m i , 2 i/i_j_n 



