56 
MR. T. Y. BAKER AND PROF. L. N. G. FI LON: LONGITUDINAL 
when we use tan /3 2n as argument, and 
B-C = —i(l—M 2 )+icZA/dM 3 . ..(IF) 
when we use sin y 2n as argument. 
That these relations hold good for the single refracting surface is readily verified 
from equations (22), (23), (25), (27) and (32). Suppose now that, for systems 1 and 3 
separately, the relation 
B-C = <r(l-M 2 )+i-dA/(M 
holds, where <r = f or — g- according to the nature of the argument, then from (48) 
and (51) 
b 13 -c 13 = B 3 -C 3 +M 3 2 (B 1 -C 1 )+A 1 M 3 M 3 2 / 1 //3 
= 0- (1 -M 13 2 )+i dA 3 /dM 3 +iM, 2 (c?A 1 /dM 1 + 4A 1 M 3 / ] // 3 ),. . (57) 
and from (47) 
f ls dAJdM 13 =f, d A 3 /dM 13 +/i{M 3 2 M 3 2 dAjdM^+A, d (M 3 2 M 3 2 )/dM 13 |. 
But 
= (fjf a ) d M, = (/,//„) M/rfM,. 
Hence 
dA 13 /dM 13 = dA.JdM 3 + M,; dAJdM. + ifAJf) d (M 3 2 M 3 2 )/dM 3 , 
and since M 3 /M 3 = const., the last differential coefficient is 4M 3 M 3 2 . 
Using this result (57) becomes 
B 13 —C 13 - <r (1 - M 13 2 ) + i dAjdM n , 
which is of the same form as the equation we started from. Hence, if the two 
components of the compound system satisfy the second invariant relation, the 
resultant system also satisfies it. But we have seen that the relation holds good for 
single refracting surfaces—hence it holds good universally. 
It should be noted that the second invariant relation is really a first order relation 
and connects the first order aberration of the inclination of a ray, with the first order 
longitudinal spherical aberration. 
§10. The Constants A, B, C, E for an Optical System Reversed and for Negative 
Lenses. 
Certain important general relations are found to hold between the constants 
A, B, C, E for rays going through an optical system and the corresponding constants 
A', B', C', E', for the same system reversed, and by making use of them we can obtain 
either set from the other. 
We arrive most simply at these relations as follows :—If after traversing the 
system we retrace our steps, the result is equivalent to compounding the system with 
