in Relation to the Coma of Optical Systems. 363 



Considering the optical paths of neighbouring rays, we 

 have the optical lengths 



0,PiQ,E, = (^PfQA, 

 QAQ1F1 = O s P 2 Q 2 F„ 



i.e., 1 P l -0 2 P ] + (Q 1 I 1 -Q 1 J 1 )-(E 1 I 1 -E 1 I 2 )+(F 1 J 1 -F 1 J 2 ) 

 = (0 1 P 2 -O 2 P 2 )+(Q 2 I 2 -QJ 8 ), 



or (0 1 P l -0. 2 F 1 )-(O l P 2 --0. 2 ¥,) + (QJ 1 -Q l J 1 )-(QJ,-Q 2 J 2 ) 

 = (E 1 I 1 -E 1 L)-(F,J,-F I J 2 ). 



We denote the angles between the axis and the rays 



O1P1, 2 P 2 , OiN and O^, 2 P 2 , 2 N by a x , a 2 ,a z and ft, ft, ft 



respectively; and the angles the axis makes with 



QJi, Q2I2, Q3I3, and QiJ,, QgJg, Q S J 3 by I 1? I 2 , T 3 and J l5 J 2 , J 3 



respectively. 

 Then 



0^-o.p, = -oA ^fr^r - -0,0, sin ab 



provided Ofi? be small. 

 ►Similarly 



QJi — QA = — IA sinlx, 

 Q 2 I 2 -QA = -IX sin L, 



FT FT- T T c OsL> — cosli 



sm(J 2 — JJ 



cosL + Qo (eosfl. + Q,)) 

 1 sin {(1,-1,; + (Q.-Q,);-' 



If now we can assume the angles Q 2 and Q, small, — that 

 is, J1J1 and LJ 2 small, we have 



FT FT- T T / cosls-coslx) f n n N cos(I 2 — Ii ) 



FA-FA- -JA{ 8in(Ii-l2) ri i-(Q,-Q0 gin(Ii-r2) 



__ Q 2 sinL — Q t sinI L ) 

 cosI 2 — coslj J 



- —TJ C0S I-' — cosT t _ Q t cosI 2 + Qo coslj 



' : 8 sinCIx-I,) ' r2 (c^s"l,-eo, 10(1 + 008(12-10). 



