54 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
using (52), and by reasoning similar to the one given for A 13 the last expression is a 
cubic in m 13 . 
Hence B ]3 is a cubic in M, 3 . 
Consider now A 13 —M 13 B 13 . This is found, after some reductions, and using (52), 
to be 
(A,-M a B,) + M s Mj a (A, -M,Bj). 
Of the above terms, A 3 —M 3 B 3 is a cubic in M 3 and therefore also a cubic in m 13 . b 3 
is a cubic in M 13 . A x —MjBi is a cubic in Mj and when multiplied by M 3 M 3 2 becomes a 
cubic in m 13 . 
Hence if the condition (I) holds good for the components, it also holds good for the 
resultant system. But we have seen that it holds for a single refracting system; 
thus it holds for any combination. Also B will be a cubic in M for any system. 
As regards C, examination of (51), remembering that for a single surface C 3 and Cj 
are quadratics in M 3 , M t respectively, leads immediately to the conclusion that C is a 
quadratic in M for any system. 
We now come to the coefficient E. Here the single refracting surface gives no 
precedent for and E 3 . Let us examine the other terms in E 13 . These can be 
written in the form /jAjM./Mg 2 (B 3 — cZA 3 /cZM 3 ) + 3A 3 M 3 2 (/ 3 B 1 +/ 1 M 3 A 1 )—2/ 3 A 3 M 3 2 C l5 
and, using A x — BjMj = Uj; A 3 —B 3 M 3 = U 3 , where U l5 U 3 are then cubics in Mj, M 3 
respectively, this is found to reduce to 
-/MW (M 3 dBJdM 3 + dUJdM*) + 3A 3 M 3 2 M 3 / (U, +f,B 1 /f n ) - 2/MU (56) 
Now 
A 1 M 3 2 M 3 3 = quartic in M 13 . 
M 3 dB jdM;, + d\J 3 /dM 3 = cubic in M 3 = cubic in M 13 . 
M 3 2 M 3 (U 1 + t /’ 3 B I //’ 13 ) = cubic in M 3 = cubic in M 13 . 
A 3 = quartic in M 13 . 
M 3 2 C\ = quadratic in M 13 . 
Hence the three terms in (56) are of form 
(quartic) (cubic) + (quartic) (cubic) + (quartic) (quadratic), 
and this leads to a rational integral polynomial of degree 7 in M 13 . 
Further consideration, however, shows that it is of degree 7 only in appearance, for 
the terms which can lead to expressions of degree 7 in M 13 are clearly 
-/ l M 3 a M 3 3 A 1 dB^M 3 + 3A s /,M : a M,(U 1 +/ 3 B 1 // 13 ), 
