PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 149 
proved that series can be found which proceed in ascending powers of the 
quantities /p, A,e™, e-1, and ev, and satisfy formally the differential 
equations which must be satisfied by the coordinates in an asymptotic 
solution ; but that these series are not convergent. 
The series in question belong, in fact, to that important class of 
developments which are now called Asymptotic Expansions ; of which 
the best-known examples are Stirling’s series for the ['-function, 
r@)= omieiie At 
and the so-called ‘semiconvergent’ expansions for the Bessel functions 
and Riemann’s -function. An example given by M. Poincaré is the 
function 
F Ww — . wu — 
( » #) DF +n 
This series converges uniformly when p is positive and|w| <w, where 
Wy is a positive quantity less than unity. If F (w, ») is developed in 
ascending powers of p, it becomes 
Dour —n)Pp. 
This series does not converge, but is an asymptotic expansion ; that 
is to say, if ¢, denote the sum of those terms for which the index of p is 
not greater than p, the quantity 
F(w, 1) =, 
a oer Ek 
tends to zero as p takes a sequence of positive values tending to zero. 
The series thus represents the function F(w, ,) for small values of p in 
the same way as Stirling’s series represents the I-function for large 
values of z. The series found in this section are of the same character, 
regarded as functions of »/ for small values of p. 
Passing in the second part of the memoir to the special discussion of a 
dynamical system with two degrees of freedom, the author studies the 
asymptotic surfaces, which have already been defined. An infinite 
number of doubly asymptotic solutions is shown to exist ; these belong at 
the same time to both of the classes of asymptotic solutions, 7.e. they are 
approximately periodic when t= —oo and t=co, but are not periodic in 
the meantime. 
Poincaré next discusses periodic solutions of the second kind. Suppose 
that a set of periodic solutions of the kind already discussed is known, 
the expressions for the coordinates being expansible in ascending powers 
of px. Let po be a definite value of ». In certain cases there exists a set 
of periodic solutions, in which the expressions for the coordinates are 
expansible in ascending powers of (u—j1)’. These are called periodic 
solutions of the second kind. Their period is approximately a multiple of 
the period of the solution from which we started. When ;: >, there are 
two of these solutions of the second kind corresponding to each value of j ; 
when p=p they coalesce into a single solution of the first kind, namely, 
the member for p= , of the set of solutions from which we started ; when 
<p they are imaginary, 
