Voiv. 7, 1921 
PHYSICS: E. C. KEMBLE 
287 
It does not seem worth while to attempt an exact discussion of the 
boundaries of the region on the complex q plane from which the zeros of 
f(q)/(q—d) must be excluded, but we can say qualitatively that there is 
little chance that the series (7) will not converge throughout the interval 
of integration if the series 
w = 2a + 3R + 4t£ 2 + 55£ 3 + . . . (21) 
converges rapidly. 
In the applications of this analysis an expression for H as a function of 
J will generally be desired. The power series may be reversed to ad- 
vantage by the following scheme which resembles that of Lagrange. Let 
H = F(J). Then by Taylor's theorem 
7-2 73 /// COO") 
H = /F'(0)+^F''(0)+-F (0) + .... M 
Let b n denote the coefficient of H n in (20). F\0), F"(0), etc., can be 
calculated in terms of the 6's. Let us first compute the derivatives of 
F in terms of H. Let 
00 
p(H) = 2« & » H "" 1= W 
.71 — \ 
Then 
fr -F'U) = T7m (23) 
dJ c - p(H) 
Differentiating again, 
F ' ^ ~ dH\p ) dJ ~ps 
S(p')' 
■pint T\ = ±(_A™ = _.il 
r UJ dH\ pzjdJ pi 
+ 
(24) 
Since J vanishes when H does, the values of F f (0), F"(0), etc., are ob- 
tained by setting H equal to zero in the right-hand members of (23) and 
(24). Thus 
F'(0) = F"(0) = -2b 2 /bl; 
F"'(0) = -663/61 + 126|/6?; 
(25) 
If these formulas for F'(0), F"(0), etc., are evaluated in terms of the ex- 
pressions for the 6's given by (20), equation (22) becomes 
r , 3 r 5/3 2 ~i T9 , 1 35 
H= — J r 
3 r 5^ 2 n 1 r 
