338 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
solutions required. The roots are |-±|- (l — l/s)*, 0, or writing, for shortness, 
<r = (l —1 /$)*, the three values of x are 
It appears that the first two solutions correspond to the complementary function 
of equation (3.613) with the term in £ neglected, so that r\ is large compared with £ 
If 5 < 1, it is imaginary, the motion is unstable and the solution fails. In the third 
solution l is large compared with q, and a first approximation to it gives a constant 
value to £ obtained by neglecting the term in in equation (3.612). It is convenient 
to obtain the first two solutions independently by a special method. 
We first omit the term in £ in equation (3.613); it is not required till the second 
approximation. Write the equation, for simplicity, in the form 
(3.623) J'-iAJ-Br, = 0, 
where 
A = Q + ih + iic, 
B — I2 2 /4 s* + iQ (k — y)— Hk — k — kc'/c. 
Remove the second term by substituting 
tj — y exp | A cfc j > 
giving 
(3.6231) y"+Afy = 0, 
where 
M - iA 2 -B+^A' 
= JQV jl + Jt (ft—K+ 2 y+N'/N) + o(^)} • 
Substitute y — Re ±,p , so that (3.6231) becomes 
(3.6232) R"± (2^P'R / + ^P // R) -P /2 R + MR = 0. 
We may make P and R satisfy any single relation we choose, e.g., 
2P'R' + F , R = 0, 
giving P ; = l/R 2 , # so that (3.6232) becomes 
(3.6233) R"—1/R 3 +MR = 0. 
* More generally P' = a/R 2 , where a is a constant, but the value of this constant is immaterial, as it 
disappears in the result. 
