MR. T. Y. BAKER AND PROF. L. N. G. FILON : LONGITUDINAL 
66 
the system reversed, and noting that the accented coefficients all refer to zero 
magnification, we have, using tangents 
tan A = tan ft', (1 + B' 0 tan 2 ft'ft) I { 1 + C' 0 tan 2 /3 f 2 ) 
and 
— F„I 0 = fn 0 (A' 0 tan 2 ft' 2 + JL' 0 tan 4 /3' 2 )f( 1 +B' 0 tan 2 ft'.ft). 
Here the suffixes in the A, B, C, &c., A', B', C', &c., have the same meaning as 
in | 11. 
Thus ‘ 
— 1 /M = (A' 0 tan 2 ft', + E' 0 tan 4 /3' 2 )/( 1 + B' 0 tan 2 ft'ft), 
whence, developing cot 2 /3' 2 in descending powers of M and stopping at the second 
term 
cot 2 ft' 2 = -A' 6 M-B' 0 + E' 0 /A' 0 . 
Substituting into 
cot 2 A = cot 2 /3' 2 + 2 (C'o-B'o) 
which is valid to the same order of approximation, we obtain 
cot 2 A = -A' 0 M + 2 (C' 0 —B' 0 ) — B' 0 + E'o/A'o.(67) 
as the second approximation for the singular inclination when M is large, the first 
approximation being cot 2 A = — A' 0 M. 
To the same order the convergency factor is . 
1+ ta,r« 0 (A'.M + {2(B',-C',) +B'„- (E' 0 /A'„)}), 
1* (%* ton*/%/«,»){A'.M»+ (8B'.-2C',-'EVA'.)M > }, . . . . (68) 
ft 2 referring to a ra 3 ^ passing through the system in the standard sense. 
Now, using the equations (HI), (64), (IV) and (V) of § 10 and equating suitable 
coefficients, we find that 
A' 0 = {n ft Inft) A 4 , E'o = {nft/nft) E e , 
C' 0 = [nft/nft) C 3 -4<r, B' 0 = (nft/nft) (A, + 2C 3 -B 2 ) -4<r 
3A a = 4B 2 -4C 2 +4 {nft/nft) a, A, = B,, 
whence, after substitution, (68) becomes 
1 + tan 2 ft, (B 3 M 3 + {B 2 -E 6 /A 4 } M 2 ).(69) 
Now, if our B leads to a sound approximation to the convergency factor for M large, 
this should be 
I + B tan 2 ft 2 , 
