100 
PHILOSOPHICAL SOCIETY OP WASHINGTON. 
Mr. Kummell remarked that the supposed analogy between the 
duplication of a cube and the trisection of an angle does not exist, 
since the latter requires a cubic with three real roots, while the 
former requires a cubic having one real and two imaginary roots. 
The subject was further discussed briefly by the Chairman and 
Mr. Bates. 
B2d Meeting. October 19, 1887. 
The Chairman presided. 
Present, sixteen members and one guest. 
Mr. G. AV. Hill read a paper on 
THE INTEGRATION OF DIFFERENTIAL EQUATIONS ADMITTING PE¬ 
RIODIC INTEGRALS. 
[Abstract.] 
The independent variable being conceived as time, a system of 
ordinary differential equations may be said to admit periodic inte¬ 
grals when the values of the dependent variables, either exactly or 
with an approximate tendency, after a certain lapse of time, repeat 
their series of values. In the latter case the longer the lapse is 
made the more nearly is the repetition brought about. Strange as 
it may seem, this subject, except in the case of simply periodic inte¬ 
grals, is, at present, not completely understood. The text-books on 
differential equations are almost wholly engaged with the cases, 
which, by certain artifices, can be integrated in finite terms or re¬ 
duced to quadratures. In the treatment of physical problems, 
however, we seldom meet with equations of this class. Far more 
frequently it is found that methods of approximation must be re¬ 
sorted to. Cauchy appears to be the author who has done the most 
for the elucidation of this part of the subject. His memoirs are in 
his later Exercises and in the volumes of the Comptes Bendus for 
1856 and 1857. 
In this paper the mode in which simply periodic integrals arise 
was discussed, and the theory afterwards illustrated by treating the 
following problem: 
Find the conditions of motion of any number of material points mov- 
