149 



jp/a (22) 



since <llp. * a^ * a?) h? Is less than unity for a < a^^. we can expand the second factor 

 in the denominator of the integrand in (19), and obtain 



r- I , 7-^ ■( ' - -^ (a + a^ * a^) * -^ (a * a' * a^) * ... . 

 / 2 I / (a - l) I 2aj^ aij 



(23) 



When Q - 0, a, - o" and the above reduces to the first term, which is Lamb's result, 

 the general case it will be seen that x can be evaluated in terms of integrals of the type 



^Ijfc- . 2 (a- 1)^2 P (a- 1). 

 J / (a - 1) ^ 



where 



P^ (z) ■= 1 * -5 Q2 + CTT ''(<' - 1) 2^ ♦ ♦ -jg^-i ^'' (2*) 



The series for x obtained in this way is convergent when 1 .f a ^ o.^f and 's useful for calculation 

 over the greater part of this range. When a = a^ the series is divergent, and to obtain values 

 of X in this neighbourhood we first consider 



"■" " ^v^ 2 /{a-^ - a-" + aj^ (a'^ _ i)} ' 



iffhich, on putting y = ^^' reduces to 



1 ( c.,^'[ g, d g 



On taking out a factor/(l - y^) from the denominator and expanding the remaining factor Dy the 

 binomial theo'em, we obtain 



x-x=/a3/2„[ y£l.L,l„3 -<. ^ ,*... Idy (25) 



A ' "> /(I- y^) I ^ ' "■ yVT^V I 



"m 



which is equivalent to the result obtained by Herring, and may be eva>uated in terms of the incomplete 

 Bet»-f unctions. Now both the expressions (23) and (25) are found sufficiently convergent for 

 calculation over a conmon range of values fora. Hence by tanlng a suitable value of a, say 

 a =0', In this range, x may be calculated from the sun of the two expressicns (23) and (25); x can 

 then be found from x » x^^ - (x - x) for other values of a > a". 



The abo.'e formulae are especially -useful when P/Q is large or moderate. If P is nearly 

 equal to 3 t'he pulsation should reduce to a simple harmonic one of small amplitude. On writing 



P • 0(1 +e). a^ = ' * ^m' f small (26) 



it is easily seen from (20) that ^^ = ^e. Substituting (l ♦ f) fora in (l9), and neglecting 

 terms beyond the first order, we obtain 



,. if II _ J,,„4!#.,l ..1 



Jo A(|)-(#-$)^ '1 i-^ J M 



so that 



a = 1 ♦ f " I ♦ f (l- cos 2x). (27) 



It will 



