284 
PHYSICS: E. C. KEMBLE 
Proc. N. A. S. 
velops f(q) into a power series about its maximum point, q = d. Let 
£ = q — d, and let H denote the maximum value of f(q). Then /(g) can 
be thrown into the form 
/(<?) = H-a? + «p + • • .)• (4) 
Introducing the symbols 
AT/ = /3£ 3 + 7 £ 4 + 5£ 5 + 
we obtain 
AF 1 (AV) 2 
2 Vtf-F 8 ^{H-vy 
Tank in effect integrates this series term by term between the limits 
- GET/a) and + (H/ a) and identifies twice the sum of the series so obtained 
with J. This procedure is wrong, since the correct limits of integration 
for J are £ = a — d and £ = b — d. Moreover, the expansion is not usually 
convergent throughout the interval of integration, so that it is not possible 
to correct Tank's work by altering the limits. 
Another scheme of series development may be suggested, which avoids 
the above difficulty. Let the quantity u be defined by the equation 
H = Vfl- /(g) = £ V« + i8 $ + 7€ * + 5 € »+.... (5) 
The sign of u is to be the same as that of £. The integral / can now be 
thrown into the form 
J = 2U H -u^du (6) 
J du 
Let us assume that d%/ du can be developed into a power series in u. Thus 
00 
1 = 2 w 
Let f+Vl 
K T = 2 1 tt * ^ H ~u*du. (8) 
J-ViT 
To evaluate if T we introduce the variable of integration 0 defined by the 
relation w = # 1/2 sin 6. 
Then (8) becomes 
K T = H 2 \ sin* 0 co$"-0 dO. (9) 
