70 
ME. T. Y. BAKER AND PROF. L. N. G. FILON : LONGITUDINAL 
Applying the equations of combination a second time, and picking out the terms 
involving A 3 , B 3 , C 3 , E 3 , we find these to be 
/ 3 E 3 M 5 2 M/ + M 5 2 M 5 * [/iA,M 3 “M 3 2 (3A 3 /M 3 —3B 3 +4C 3 ) +/?A 3 M 3 2 (36,-20,)] 
+/ 3 A 3 M 5 2 M 5 2 [3A 5 /M 5 - 3B 5 + 4C 5 + 4a- (1 - M- 2 )] 
+ 3B 3 / 5 A 5 M 3 2 -2C 3 / 5 A 5 M 5 2 . 
Hence the lens 3 contributes to the final E 
/ 3 A 3 {3M 5 M 5 2 [A b + ( flf,) AAB,M 35 2 ] + 4o-(1-M 5 2 ) M 5 2 M 5 2 
+ M 5 2 M 5 " [ — 3B 5 + 4C 5 + (3B, — 2C,) M“ 35 ]} 
+ B 3 M 5 2 (3/ 5 A 5 —3/ l A 1 M 35 2 M 35 2 ) 
+ C,M., 2 (-2/ 5 A 5 + 4/ A,MHM ?5 2 ) 
+/«%*, 
and the A’s, B’s and C’s in the curled brackets can be expressed in terms of the 
individual lenses of the system by means of equations (71), (72), (73). 
If we denote the coefficients of A 3 , B 3 , C 3 , E 3 , in the above by l 3 , m 3 , p 3 , q 3 , then the 
contribution of the individual lens to E 
— 4A 3 +m 3 B 3 +_p 3 C 3 +g 3 E 3 . 
Hence, if we vary K 3 for this lens, keeping focal length and magnifications 
unaltered 
AE = AK 3 (4 0A ? /3K 3 + m 3 dB 3 /dK 3 +p 3 0C 3 /0K 3 + g 3 SE 3 /SK 3 ). 
If all the lenses are simultaneously varied, then we have 
AE = 2 AK (l 3 A/3K + m dB/dK + p dC/dK + q 3E/3K). 
We have similar equations for A A, AB, AC, but they take a much simpler form. 
Using these, we can, if we have enough lenses, vary the K’s so that, between limits, 
we can make our four constants A, B, C, E take up any assigned values, or, if we 
wish to keep any one constant whilst slightly varying the others, we have a linear 
relation between the AK’s. 
§ 16. Conclusion. 
We have now established a formula of fractional type for the longitudinal aberration 
of a symmetrical system which, while algebraically correct as far as the second order, 
does in fact, give results beyond this order in those numerical cases which have been 
tried, and largely overcomes the difficulties of slow convergency in critical regions. 
