PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 145 
certain conditions the motion in the restricted problem of three bodies is, 
in this sense, stable,—or stability ' may be taken (as by Poisson) to mean 
that the system is to pass infinitely often through positions as near as we 
please to the initial position ; the intervening oscillations may be of any 
magnitude. 
The existence of asymptotic solutions (which will be explained later) 
shows that an infinite number of particular solutions of the restricted 
problem of three bodies exist, which are not stable in Poisson’s sense of the 
word. But M. Poincaré now proves that there are also an infinite number 
which are stable, and, further, that the former are the exception and the 
latter are the rule, in the same sense as commensurable numbers are 
the exception and incommensurable numbers are the rule. In other 
words, the probability that the initial circumstances may be such as to 
give rise to an unstable solution is zero. 
The proof of this is made to depend on the following theorem ; when 
the point (x, 2, .- + 2ny Yiy Yo +++ Yn) Moves so that its coordinates are 
always finite, and the integral invariant 
[aa re A eee ie 
exists, then for every region 7) in space, however small this region may be, 
there exist trajectories which pass through 7, an infinite number of times ; 
and, in fact, those points of 7, which do not give rise to such trajectories 
form a volume which is infinitely small compared with 7». 
It is thus shown that, when the constant of relative energy in the 
restricted problem of three bodies lies between certain limits, the motion 
is stable not only in the sense of Hill and Bohlin, but in the sense of 
Poisson ; the number of exceptional cases to which this law does not 
apply being infinitely small in comparison with the number of orthodox 
cases. The result is, however, not extended to the general problem of 
three bodies. 
The author next passes to the theory of periodic solutions. 
Consider a system of differential equations 
div, ° 
aoe G=lI, 2; eee 2); 
where the X’s are functions of ~,, %,...,, and a parameter p; 
X,, X,,...X, may also involve ¢, but if so they are supposed to be 
periodic functions of ¢ with a period 27. 
A periodic solution of these equations of period T is of the form 
x=¢(t), (i=1, 2,... 7) 
where the functions ¢ are such that ¢(¢+T)=¢,((t). (If x, is an angular 
variable, this may become ¢,(¢+T)=9,(¢) +27, where 7 is an integer.) 
The meaning of this for our purpose is, of course, that the relative 
motion of the moving bodies repeats itself at regular intervals T of time. 
Suppose that, for the value 0 of the parameter ,, these equations 
admit of a periodic solution, 
Hi—9,(t)y (G1, 2, . she 90) 
1 A discussion of general definitions of stability is given in the second volume of 
Klein and Sommerfeld’s Theorie des Kreisels. 
. L 
