288 
PHYSICS: E. C. KEMBLE 
Proc. N. A. S. 
2« + T^~^^J J + (26) 
As a check on the series development here suggested, the writer has 
derived Sommerf eld's formula, 
* <? <z 2 \_<ac J 
(27) 
by means of the series (12). The derivatives of \p when evaluated by- 
equations (16) are in this case particularly simple and as a result the series 
can be summed. 
It is of interest to note that the method of development in series here 
suggested is applicable to a variety of problems. It may be used to 
evaluate indefinite as well as definite integrals. 
One simple application is in the determination of the periods of os- 
cillation of a vibrating system. Iu the case of a conditionally periodic 
system with orthogonal coordinates, the periods are given by expressions 
of the form 
<!$ = 9 C b dg 
V/5) 
T=f d ^ = 2 P 
(28) 
where a and b are again roots of f(q). Introducing the quantities H, £, u 
defined as in the preceding work, we obtain 
dt/du 
7= : du. (29) 
If d£/du is developed into a power series of the type (7), we obtain 
T 
where 
^ 2 yn =M 
The coefficients K T vanish when r is odd, and the final expression for the 
period is 
00 
T = 2t [ a a + 2^^, (r+1) V (2r) H j . (30) 
r=l 
1 This is in effect the method used by Sommerfeld, Atombau und Spektrallinien, 
2nd edition, Braunschweig, 1921, pp. 476-482. 
2 F. Tank, Mitteilungen Physik. Ges. Zurich, 1919, p. 87. 
3 Cf. G. Hettner, Zs. Physik., 1, 4, 1920 (350). 
