48 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
The cases are discriminated by the vanishing of these square roots, which occurs 
when cos \Js 0 or cos \[s 2 = 0. 
It appears, therefore, that the vanishing of the factors in the denominator of (38) is 
wholly irrelevant, and, if we adopted for B the value given on the right-hand side 
of that equation, we should thereby be introducing, in the neighbourhood of Mj = 
(w 0 + 7i 2 )/2n 0 , (n 0 + n 2 )/2n 2 entirely irrelevant singularities, which would make the 
formula worthless. 
The question arises, what is the range of values of IVb for which the equation 
4n 0 2 sin 2 x/n 2 2 = — R 
is valid and legitimate ? 
If we start from = oo, which corresponds to a real case, the signs of the square 
roots in (39) are well determined, and the correspondence between sin 3 X and is 
unique and definite and can be continued until we reach a point where one case 
passes into another. These cases we have found to be the branch-points of the 
three square roots, namely :— 
u = 0, cos \{s 0 = 0 and cos \fs 2 = 0. 
u = 0 leads to x 0 = cc or = 0.(A) 
cos = 0 leads to sin 2 X = u, or, using the first form of (34) 
4 n 2 ujn 2 = 4w—(I + u—n 2 /n 2 ) 2 , 
i.e., (l —u—n 2 /n 2 2 ) 2 = 0, that is u = l—n 2 /n 2 , leading to 
x 0 = ±r x (l-n 0 2 /n 2 2 )~* and ^ = {l±{n 2 -n 0 )/^/(n 2 2 -n 0 2 )}~ 1 . . . (B) 
cos \fs 2 = 0 leads to sin 2 X = n 2 u/n 0 2 , that is, to 
u = n 2 /n 2 2 - 1, x 0 = ±r x ( )“*, M, = { l±{n 2 -n 0 )/^(n 0 a -n 2 2 )}~ 1 . . (C) 
If n 0 > n 2 , both values of Mj given by (B) are imaginary. The values given by (C) 
are both positive, M t = (l +(n 2 —n i) )/^ s /(n 0 2 —7i 2 )}~' 1 being the greater. 
The range over which we can travel without ambiguity is, therefore, from 
M, = + cc to I, = {l +(n 2 —n 0 )/ x /(n 2 —n 2 )}- 1 and from = — co to Mj = 0. 
If n ti < n 2> the values of HS. 1 given by (C) are imaginary, those given by (B) are 
positive, and M, = (l—(w 2 —n 0 )/^/(n 2 2 —u, 0 2 )} -1 is the greater, so that the range of 
validity is from = + oo to Mj = {l —(n 2 —?i i) )/ x /(n 2 —n 2 )}~ 1 and from M, = — oo to 
= 0 . 
Within this range (l +4w 0 2 sin 2 a 0 /n 2 2 R) is the correct convergency factor ; outside 
this range it is irrelevant. 
It is clear, then, that we cannot find a single formula for the convergency factor, 
which will hold for all values of the magnification. 
