PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 7 OCTOBER 15, 1921 Number 10 
THE EVALUATION OF QUANTUM INTEGRALS 
By Edwin C. Kemble 
Jefferson Physical Laboratory, Harvard University 
Communicated by E. H. Hall, April 22, 1921 
The application of the Wilson-Sommerfeld quantum conditions to a 
conditionally periodic system with orthogonal coordinates involves the 
evaluation of an integral of the type 
J = £ <Wdq. (1) 
The integral is to be extended over a. complete cycle of values of q, 
which oscillates between two roots of f(q). The sign of the radical is to 
be the same as that of dq, so that if a and b denote the roots of f(q), the 
integral can be written 
J = 2 fyW~dq (2) 
If f(q) is a polynomial of the second degree in either q or l/q the integral 
can be cleanly evaluated. Otherwise, approximations are generally 
necessary. If f(q) can be expressed in the form 
f(q) = <p(q) + ol f(q) 
where <p(q) is quadratic in q or l/q, a is constant, and oafr(q) is small, a 
natural method of procedure is to try to develop / into a power series in 
a. Thus 
/(«) = 7(0) (0) + ^ + ^ +..•• ® 
7(0) and J r (0) are easily evaluated, but unfortunately the higher deriva- 
tives of J with respect to a cannot be calculated by the usual methods be- 
cause the higher derivatives of V (fq) with respect to a become infinite 
at q — a and q = b. Hence this method is useful only when the higher 
order terms are negligible. 1 
Another method of attack employed by F. Tank 2 and accepted as 
valid by others 3 turns out on close examination to be faulty. Tank de- 
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