PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 147 
of ¢, and to be periodic in the y’s of period 27. F is further supposed to 
depend on an arbitrary parameter p, and to be expansible in the form 
FEFot+pF+ pF, + F3 +... 
where F,) does not involve the quantities ¥. 
When p=0, the equations can be integrated ; x), », «3 are in this case 
constant, and 
y=nti+w,, where n,=——? 
0a 
the quantities w, being arbitrary constants of integration. The solution 
will be periodic if m,, ”,, 3 are commensurable with each other ; the 
period T will then be the least common multiple of the quantities 
oF, 
“4 
Qn Qe Dr 
eee 
N, Ny Ne 
In general, it will be possible to choose 2, %, #3 so that 7), no, 5 
may have any prescribed values—at least in some domain ; so that there 
are co? periodic solutions, when p is zero. 
The author next proceeds to investigate whether periodic solutions of 
period T exist, when « is not zero. By a process of reasoning similar to 
that used before, it is shown that, corresponding to any periodic solution 
which exists when »=0, and whose constants satisfy certain conditions, 
there exists in general a solution of period T when p has small values dif- 
ferent from zero. The quantities x, and y;—,t can be expressed as series 
of ascending powers of p, the coefficients in which are circular functions of 
Int 
pT and a method of forming these expansions is given. 
The results are applied to the restricted problem of three bodies ; a 
difficulty arises, which in this case is solved by asimple artifice, but which 
is not so easily disposed of in the general problem of three bodies. 
Still considering the dynamical system with three degrees of freedom, 
Poincaré considers a solution 
U=P(t)+E, Yi=v(t)+n, 
differing but little from a periodic solution, and writes 
E=e'S;,, ne 'T,, 
where §; and T; are periodic functions of ¢. 
When the periodic solution corresponds to »=0, the six exponents a 
are all zero ; when p is not zero it is shown that the quantities a, S,, and 
T; are expansible in ascending powers (not of p, but) of Wp. It is shown 
that to every set of values of m, and n,. there correspond at least one 
stable and one unstable periodic solution ; and that there are exactly as 
many stable solutions as unstable when , is sufficiently small. 
The next idea to be introduced is that of asymptotic solutions. Re- 
turning to the general system 
BX, Gest Fem) 
let Ate 
v t 
L2 
