286 
PHYSICS: E. C. KEMBLE 
Proc. N. A. S. 
,,/ * 2v' 4vw' 
4Vy 
4v'w' 2vw' 
w 
w 
4, "'(«)=- \ W + 2m' " _ - U ™"«>1 _ 4 (v Tw' _ Uw'w' 
w w 
+ 
50vv'(w') 2 40 v 2 w'w" 4v 2 w"' 60v 2 (w')< 
w 2 
(17) 
At the point 
« = £ = (), 
w = 2v = 2a, w' = Sv' = 3/3 
w >'=4 v » = 8y 
w (n) = ( n + 2) 1> (m) = 
(18) 
Hence 
*(0) = 
*'(0) = 
1 
a 5 / 2 
{If-} 
r a' A \ 16 a 3 2 a« 1 a a 7 
(19) 
Combining the above equations with (13), (11), and (12), we obtain 
+ 2/y r 
This result is similar in form to Tank's and the coefficients of the two 
lowest powers of H are the same as in his series, but the coefficient of H* 
is different. 
In order that the result may be valid, it is necessary that the series (4) 
converge and represent f(q) throughout the interval of integration, and 
that the derivatives of \f/ shall all be finite throughout a circle whose radius 
is greater than -\-H^ 2 drawn about the origin on the complex u plane. From 
(17) it is evident that the derivatives of \p will be finite up to the point 
where w vanishes. It follows from (14/) that f{q)/{q — d) must have no 
zeros on that part of the complex q plane which is mapped on the above- 
mentioned circle on the u plane. It is necessary, in particular, that 
f'(q) shall vanish not more than once for real values of q between a and b. 
