MATHEMATICS: W. D. MACMILLAN 
437 
For testing the importance of the lymphoid tissue in natural immunity 
we can only compare the percentage of takes in X-rayed and normal 
animals inoculated with the same cancer. This has been done with a 
variety of different cancers and a large series of animals. The average 
number of takes in the X-rayed animals was 94%, while in the untreated 
animals only 32% of those inoculated grew the cancer. This shows a 
very considerable destruction of the natural immunity accompanying a 
destruction of the lymphocytes. 
To summarize, we have shown that a marked increase in the circulat- 
ing lymphocytes occurs after cancer inoculation in mice with either a 
natural or induced immunity. When this lymphoid reaction is pre- 
vented by a previous destruction of the lymphoid tissue with X-ray the 
immune states are destroyed. Hence it would seem fair to conclude that 
the lymphocyte is a necessary factor in cancer immunity. 
SOME THEOREMS CONNECTED WITH IRRATIONAL NUMBERS 
By William Duncan MacMillan 
DEPARTMENT OF ASTRONOMY. UNIVERSITY OF CHICAGO 
Presented to the Academy, May 27, 191 5 
As is well known to those who have investigated the fields of celestial 
mechanics, the series which arise there from the integration of the 
equations of motion involve factors of the form (i—jy) in the denomina- 
tors of the coefficients, where i andy run over the entire series of positive 
integers and 7 is a positive number which may be rational or irrational. 
Previous to the time when Poincare had shown the existence and con- 
struction of periodic solutions (in which 7 is always rational) it had 
been the custom for the astronomers to regard 7 as irrational since with 
this hypothesis the factors {i—jy) never vanish and consequently non- 
periodic terms did not arise in the solutions. The presence of these 
factors in the denominators naturally led to very grave doubts as to 
the convergence of these series since there are infinitely many such 
factors which are smaller than any assigned limit, and the convergence 
has never been proved. 
Considerations of this nature have led me recently to examine the 
convergence of simple types of power series in which this phenomenon 
occurs, and it has been found that the series 22 (Uj/{i —jy) Xi<' has 
precisely the same domain of convergence as the series SZay Xi* X2J', 
provided 7 is a positive irrational number which satisfies a rather mild 
condition, which is stated below. 
