Vol.. 8, 1922 
PHYSICS: P. S. EPSTEIN 
167 
books both of Riemann and of Sommerfeld.^ Therefore the assertion of 
Mr. Kemble that the expansion (3) is appUcable only when terms of the 
second and higher order in a are neghgible is entirely unfounded. An 
expansion of this type was first introduced into the theory of quanta by 
the present writer in his work on the Stark effect and, as a matter of fact, 
the computation was carried through by him not only for the term of the 
first order, but also for the term of the second order ^ and proved to be as 
easy as can be desired. 
The method followed by F. Tank in a paper referred to by Mr. Kemble 
is but a slight modification of this procedure. The purpose of Tank is 
to find an expansion for the integrand ^f{q) when it is not given in the 
form (2). This purpose is reached by developing /((/) in a power series 
from the point q = d, which makes /(g) a maximum, li q — d = ^ and if 
H denotes the maximum value of f{q), this function can be represented 
in the form 
fiq) = H-ae- + 7^'+ H-ae-\ 
wherefrom 
2 _ 
2 <H-ae ^{<H-aeY 
To this series the method of complex integration can be applied term- 
wise in exactly the same way as to expansion (3). Since the complex 
path of integration amounts to a real one between the roots of the radi- 
cand H — a^^, whenever this real integral is convergent. Tank was per- 
fectly justified in taking as limits —{H/aY^ and Moreover 
I cannot agree with the opinion of Mr. Kemble that * 'the expansion is not 
usually convergent throughout the interval of integration." It can be 
shown in a general way that the problem of convergence cannot lead to 
any difficulties. 
Summarizing, we can say that both methods in question are correct 
and have been successfully applied to the treatment of important prob- 
lems. The new procedure proposed by Mr. Kemble may answer its pur- 
pose as a method of computation, but it surely does not equal the old 
ones in directness and mathematical elegance. 
1 B. C. Kemble, Proc. Nat. Acad. Set., 7, 1921 (283). 
2 Paul S. Epstein, Ann. Physik., 50, 1916 (489). 
3 B. Riemann, Schwere, Elektrizitdt, Magnetismus, section 27, Hannover, 1880. 
4 A. Sommerfeld, Physik. Zs., 17, 1916 (500). 
^ A. Sommerfeld, Atombau u. Spektrallinien, pp. 476-482. Braunschweig, 1921. 
« P. S. Epstein, Ann. Physik., 51, 1916 (184). 
