40 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
which is correct to the first order when q and t are tangents and to the second order 
when q and t are sines. 
In the above A, B, C are polynomials in M of degrees 4, 3, 2 respectively, so that 
the empirical formula is well justified for the simple refracting surface. In this case, 
too, it is possible to calculate r directly, and, in fact, a simple geometrical construction 
is given for it. When this is followed for varying image-positions, it is found that 
outside a certain range of M, the r so obtained becomes irrelevant, and that, in fact, 
if the correct factor 1—£ 2 /t 2 is retained in the denominator of Ax, although it 
improves the fit by removing singularities in the range round M = co 5 it introduces 
entirely fictitious singularities in other and important parts of the range, and makes 
the formula worthless. 
A good deal of light is thrown upon the problem when it is found that, if we 
develop — in descending powers of M in the neighbourhood of M = oo 5 the two 
leading terms are discovered to be identical with the two leading terms of the 
cubic B, previously obtained. This makes our B approximate more and more closely 
to \ precisely as the effect of the denominator term becomes more important, and it 
T 
is this fact which is the key to the numerical value of the method. 
We then proceed to show how the constants for a combination of the two systems 
can be obtained from the corresponding constants of the individual systems. In 
doing this it appears that, so soon as we pass from the single refracting surface to 
the lens, a new constant is introduced into the formula, which now takes the form 
Ax 
A£ 2 + EC 
1 + B£ 2 
(7) 
where A and B are of the same form as before, but E is now a polynomial of 
degree 6 in M. In the case of a lens the term EC is found to be, in general, of 
small importance, which accounts for the good fit of the empirical formula. 
The formula (7) for a combination holds good to the second order inclusive, and B 
agrees with —, when M is large, as far as the leading term only. For numerical 
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purposes, however, a correction is discussed, which is very readily applied, and which 
makes the two leading terms in B agree with the two leading terms in as in the 
t“ 
case of the single refracting surfaces. 
The formulae for combining two systems take comparatively simple forms; the A, 
B and C for the combination are expressed as linear functions of the A’s, B’s and C’s 
of the components, and the E as a lineo-linear function of the A’s, B’s and C’s of the 
components, each term involving a product of which one factor belongs to one 
