TRANSACTIONS OF SECTION A. 4.4.7 
move in Keplerian ellipses E, and E, round their primary. At a certain moment 
their mutual distances become infinitely small, a great perturbation ensues, and 
after that the two bodies move again in Keplerian ellipses E,’ and E,’, different 
from the former. This may be repeated, and Poincaré shows that it is possible 
that after a finite number of meetings the original orbits E, and E, are repro- 
duced. The solution is then periodic, and Poincaré further shows that such a 
solution can remain periodic if the masses are made finite. The shortest distance 
then, of course, is no longer infinitely small, but only a ‘ near approach.’ 
Now an example of an orbit of this kind is offered in the work of Sir George 
Darwin. It is the orbit a2,=1:08 figured on page 169 (‘ Acta Mathematica,’ 
vol, xxi.). Darwin’s orbit starts with the two bodies in symmetrical conjunction 
in the middle of the large perturbation. At the end of the part of the orbit con- 
strued by Darwin the orbit of the smaller body P has become practically a 
Keplerian ellipse, with longitude of perihelion . (The other body, J, of course 
continues to move in its circular orbit, the mass of P being zero.) Let the time 
taken by this first part of the orbit be ¢,. The angle between P and J at the end 
of the time ¢, is then —n#t, +, being the mean motion of J. Now put 
(1) nt,=7—nt, ta+T 
where r is a small angle, the meaning of which will be explained presently. If 
it so happens that P in the time ¢, has completed a whole number of half-circuits 
in its elliptic orbit, the perihelion of this ellipse having in the same time advanced 
through the angle r, then at the time 3T =¢,+ ¢, the bodies J and P will again be 
in symmetrical conjunction or opposition, and the orbit will therefore be periodic 
with the period T. 
The condition that this is so is 
(2) n't, = ker 
where & is any integer and »’ the motion of the mean anomaly of P in its ellipse. 
Now by Darwin’s work the time ¢, and the elements of this ellipse are given 
as functions of, say, 2,. Then (2) gives ¢, as a function of n’, and therefore of «,. 
Further, an easy computation will give the perturbation of the perihelion during 
the time f,, .e., the angle 7, as a function of ¢, and of the elements of the ellipse, 
and therefore of 2,. Thus in (1) all quantities are functions of 2, and it is pos- 
sible to choose a, so that it shall be satisfied. The periodic solution of the 
second species is thus perfectly determinate. 
From Darwin’s figures I find, very roughly, 
nt, =114° w = 62° 
and for the elements of the ellipse I find 
a’ =0'37 e’ =0°69 n'|n = 4:24 
In this very rough approximation we can take r=0; we then find from (1) 
nt, = 128°; and if we take k=3 we have from (2) n’/n=4-22, 
It is thus evident that Darwin’s orbit 2, = 1°08 if not itself periodic is at least 
very near to the periodic solution of the second species for C=39-0, An exhaus- 
tive investigation would, of course, involve a considerable amount of computation, 
which I have not yet found occasion to undertake. 
4. On Essentially Positive Double Integrals and the Part which they play 
in the Theory of Integral Equations. By H. Baremay. 
§ 1. By an integral equation of the first kind we shall understand an equation 
of the form 
6b 
f=| oy OO. 2c. 1 
in which f(s) and g(s, ¢) are given for values of s lying befween c and d, and p(t) 
