58 
MR. T. Y. BAKER AND PROF. L. N. G. FILON : LONGITUDINAL 
(61) becomes 
E—M 2 M 4 E 7 +A {B + M 2 B 7 —2C —A/M —4a-(l —M 2 )} = 0. . . . (62) 
Now substitute from (60) for B + M 2 B 7 the value 
C + M 2 C'+A/M, 
and (61) leads to 
E —M 2 M 4 E' +A{M 2 C 7 —C —4o-(l—M 2 )} = 0.(63) 
For a single refracting surface, where E, E 7 are identically zero, this must lead to 
M 2 C 7 —C = 4o-(l—M 2 ).(IV) 
a result which is easily verified from equation (32). 
Now consider a system compounded of two systems. For the system direct, we 
have 
C 13 = Cg+M/Cj. 
Similarly, for the system reversed, change C 3 into Cfi, G\ into C 7 3 , M :1 into I /Mj. 
c 31 = C , 1 +If , C ' 3 
M 13 2 c 31 -C 13 = M, 2 M 3 2 C'j—M 3 2 Ci + M 3 2 C' 3 —C 3 
= M :f 2 (M 1 2 C' 1 -C 1 ) + M :i 2 C' 3 -C 3 
and using (IV) which we know to be true for a single refracting surface 
M 13 2 C 31 -C 13 = 4o- [M 3 2 (1 — Mj 2 ) +1 — M 3 2 ] 
= 4o- (l —M 13 2 ). 
In other words (IV) will hold for the resultant system if it holds for the components, 
and therefore as in previous similar cases, it holds for any system. 
We shall call (IV) the fourth invariant relation. 
Equation (63) then shows that there exists & fifth invariant relation 
E=M 2 M 4 E 7 . (V) 
or 
E (M )/n 0 2 = M 6 E 7 (M l )/nf 
so that E possesses a property similar to that of A, previously noticed, viz., if we 
divide it by the square of the initial refractive index, the coefficients of powers of M 
equidistant from the beginning and end of the development in M are interchanged. 
Equation (59) has therefore led us to two independent invariant relations. 
On the other hand it will be found that (60) leads to no new relation. For if we 
substitute into it for B 7 — C 7 and B —C in virtue of the second invariant relation, and 
then use the third relation, it becomes an identity. 
