in Relation to the Coma of Optical Systems. 365 



This result is equivalent to that proved above for the rays 

 passing through a?! = 0, on the stop, provided that for ^ = 

 the Coma may be denoted by 



i x sin 2 A -\-i 2 sin 4 A, and for x 1 =p 1 by 3^ sin 2 A + 5i 2 sin 5 A. 



This shows that the proper expansion for the Coma is in the 

 form ii sin 2 A + i 2 sin 4 A rather than z\A 2 + * 2 A 4 ,since the latter 

 would not separate the terms which are in the ratio of 3 : 1 

 from those in the ratio of 5 : 1, and thus would lead to a more 

 complicated expression for the Coma. 



The summation above may also be applied to the case of an 

 oblique ray. We then have 



— 1 2 (sina 1 — sin« 3 )— (IxJi sinl]— J^ sin I 3 ) = S(JiJ 2 — IJ2) sin I 



+ 2J 1 J 2 COS 3 I.^|, 



/ — Oi0 2 (sina 1 — sin« 3 ) \ 1 f /t T T T \ • t 



+ S((J:J 2 - IJO sin I, ) + S ((J, J 2 cos »I H) } . 



The summation may be effected provided we know the 

 approximate expressions for the quantity JiJ 2 , but in the 

 ordinary way these expressions are too complicated to admit 

 of simple expression. 



The condition for the absence of Coma for a small object 

 not on the axis is easily found provided one point be perfectly 

 defined : 



., . / sin a — sin a .\ A ,. . . 



it is( q . : . — 1 1 =0 throughout the aperture, 

 V sin 1 — sin I / 



where a refers to the ray through the centre of the stop. 



In conclusion, it will be seen that the Comatic effects asso- 

 ciated with any departure from the sine condition may be 

 calculated, even though spherical aberration be present, from 

 the relation (III.) above. 



