52 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
Remembering that t 2 — M 3 £ 4 and retaining only the first two terms of the denominator 
product 
Axjn i — (f l3 A ri t i +f 13 E 13 t i i )j(l f B 13 £ 4 2 ).(46) 
where 
JrS-^13 — ./sAy+yjAiMg 2 !!;/.. (47) 
B 13 = B 3 + B 1 M/ + A 1 M 3 M 3 2 / 1 //3 .(48) 
yi.sE 13 — / 3 E 3 +yiE 1 M 3 2 M 3 4 
+ / 4 A 1 M 3 2 {B 3 M 3 2 + A 3 M 3 —M 3 2 M 3 2 d (A,,M ; r 2 )/dM 3 } 
+ / 3 A 3 M 3 2 {3B1 — 2 Cj}.(49) 
Again 
__ g 2 (M 3 +A M 3 ) -1 (1 + ( B 3 +AB 3 ) g 2 2 (M 3 +AM 3 )- 2 } 
qt l+(C 3 +AC 3 )g 2 2 (M 3 +AM > 2 
and retaining only terms of order t 2 
g 4 = q 2 (M 3 + AM ,)- 1 (1 + B 3 trM ; r 2 )/( 1 + C 3 U 2 M 3 -°) 
= tMr 1 (l —AM. 3 /M 3 ) (1 + B : U 2 ) (1 + B ; /yM :; - 2 )/{( 1 + C^ 2 2 ) (1 + CVyM.y 2 )} 
= tMr 1 (1 + B 1 ^+B 3 i 2 2 M 3 - 2 + ^A 1 M 3 / : // 3 )/(l + C ^ 2 + C 3 * 2 2 M 3 - 2 ) 
— £ 4 (l + B 13 £ 4 -)/(l + C i3 £ 4 2 ), . ..(50) 
where B ]3 has the value given by (48) and 
C 13 = Ca + CiMa 2 ..(51) 
The equations (47), (48), (49), (51), give the constants for the combined system in 
terms of those for the components. It may appear at first sight as if the choice of 
the constants B 13 and E 13 had been arbitrary, for clearly, if X be any quantity, 
1/'i,'Ai 3 £ 4 2 +f L3 (Ei 3 + XA 13 ) 1 4 4 }/{1 + (B 13 + X) £ 4 “} 
will give a development equally valid to the second order. But, if we do this, and 
we wish to preserve the simple character of the relation (48) giving the B for the 
combination, X will have to be a linear function of A l5 A 3 , B l5 B 3 . XA 13 must then 
necessarily contain terms of one or other of the forms AjBj, A x 2 , A 3 B 3 , A 3 2 . Thus 
the new E will contain such terms and will no longer be of type (49) which is linear 
in the aberration coefficients of each system taken separately. Thus the lineo-linear 
type of equation for E 13 requires X = 0 . 
We note also that the equations of combination are identical in form, whether we 
are dealing with the sine or the tangent of the inclination as argument. 
