340 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
Similarly we have 
verifying that £ is small compared to r\. The contribution of £ to equation (3.613) is 
thus 
(3.6235) 
2kc'ij 
C (1 ±< x ) 
which is equivalent to an addition to the coefficient of >/ of terms which are 0(l/Q 3 ) 
compared to the principal term. The solution can be repeated with these terms 
included, but is unaffected to the order to which we are working. We shall take our 
first two standard solutions in the form 
(3.624) 
(3.625) 
>h 
— (SVoY /yiPl 
Qcr 
*12 
\* - tP , 
a ,r 
ii 
& = 
2fo?i 
icQ (1 + <r) 
210)2 
icQ (1 —or) 
The differential coefficients of the solutions may be obtained by differentiation of 
these equations. 
For the third solution we have shown that the exponential index is zero to the first 
order, and that a first approximation is given by 
% — £ 3 = tz — 1- 
The expansions take a somewhat different form, like those for the particular 
integral, and we write 
(3.6261) r, 3 = J 0) + r, (1) /iQ + J 2) /{iQ) 2 ... , 
(3.6262) = f to) + f w /tO +WQ)*... • 
Substituting in equations (3.612) and (3.613), we obtain 
»? (1) = 4sc' — —4s 6\ sin 6 U 
<? 1] = 4 [ Ksc'dt/c. 
Jo 
The significance of this solution will he considered after the particular integral has 
been discussed. 
Our standard third solution is then 
(3.627) 
4 sc' 
kSc' 
dt. 
