364: Mr. S. D. Chalmers on the Sine Condition 



Thus 



(HJ 1 I 1 -B 1 I i )-(F 1 J 1 -F 1 J s ) = (JiJ 2 -IiI 2 ) C ^V C T S V 



T T Qi cos I 2 + Q 2 cos I 2 u 



t* " 1" 2 i i Ft t~^\ ' 



l + COS^Ix — Ig) 



or 



— Oi0 2 (sin «!— sin <x 2 ) — (IjJx sin Ix— I 2 J 2 sin I 2 ) = (JiJ 2 — Iil 2 )— wt p3 



sm^ix— -i 2 ) I 



TT ( IiJi I 2 J 2 1 / cos I x cos I 2 \ 



+ 13 IQiJi Q 2 J 2 J U+coBfr-l,)/ 



On summing throughout the aperture for Ox on the axis 



— OiOo sinax— IxJi sinlx = 2(J"iJ 2 — Ixl 2 ) sinlx + SJ^ cos 3 Ix 2 .^ry, 



where the value o£ JjJ 3 is assumed small as compared 

 with NI. 

 Thus 



V. Sinll (^f£f - 1 ) = -(Vs-Wsinlx 



+2(JiJ a «IiI,) sinI+2(jiJ 2 cos 3 I.^. 



But IxJx — 1 3 J 3 = Comatic error for rays in one plane ; and 

 2(J 1 J 2 — Iil 2 ) sinlx is the Coma X the average value of 

 sinlx: this average value depends on the way in which the 

 Coma varies with the aperture. 



If we assume 



O T! 



— = 3ix sin 2 A + du sin 4 A, 



SO -S3 



the mean value in the first term is 



2«x sin 2 A, 

 u e., I of 3^ sin 2 A, 



in the second % of 5^ sin 4 A. 

 Thus we have, since A= — I l5 



( - T t 2 • — ir — 1 ) = tt" "1 4 Coma of first order + 1 Coma of second order > 

 \I 3 J 3 sin A x / I 3 J 3 I 3 ^ 5 ) 



, Vs 1 ") 



+ TN sinlj 



= yy \i Coma of first order 4- J Coma of second order > 



long, spher. aberr. 



+ " ~ IN 



