PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE popirs. 155 
1898-9. The first half of it is devoted to the theory of Invariant 
Integrals, which is given here in a more developed form than in the 
memoir of 1890; while the second half is concerned chiefly with the 
theory of periodic solutions of the second kind. Since the publication of 
the 1890 memoir, periodic solutions had been connected by the author! 
with the theory of least action. In the first of the two notes referred to 
it is shown that the existence of periodic solutions of different kinds can 
be inferred from the principle of least action, when the law of attraction 
is some inverse power of the distance higher than the square; in the 
second note, a classification of unstable periodic solutions is made, which 
depends on the principle of least action ; and it is shown that when the 
constants of the motion are varied, a periodic solution cannot pass from 
one kind of instability to the other. In this volume, the theory of least 
action is further applied. 
After developing the theory of periodic solutions of the second kind, 
Poincaré shows that some of the results of Darwin’s paper of 1897 are in 
accordance with his own theorems, and criticises others ; and terminates 
the book by a study of doubly-asymptotic solutions. 
Since 1892 Brown? has published several memoirs dealing with the 
junar theory on the plan projected by Hill. The first paper extends 
Hill’s paper of 1878 by including in the work the inequalities which 
involve the sun’s parallax ; in other words, Hill found periodic solutions 
of the motion of a particle in a plane under the influence of two bodies 
which revolve round each other in circular orbits, and whose distance 
apart is infinite, while Brown supposes this distance to be finite. In the 
second paper the inequalities dependent on the moon’s eccentricity are 
included, 2.e. the general solution of Hill’s problem, which of course is 
not periodic, is found. The investigations of the third paper relate to the 
more general problem of the moon’s motion, and include a deduction and 
extension of the theorems of Adams’s memoir of 1878. (See § IV.) Brown 
is at present preparing a complete numerical lunar theory. 
In 1895 Hill’ calculated numerically the periodic solution, which may 
be taken as the base of the lunar theory, and in 1896 Liapounow!? dis- 
cussed Hill’s series, and proved their convergence in the case of the actual 
motion of the moon. 
Andoyer? in 1890 gave another method for finding the solution of the 
differential equations of Dynamical Astronomy by means of series of 
periodic terms. He obtains the series directly by assuming that they are 
of the required form but with undetermined coefficients, and finds these 
coefficients by successive approximation. It is shown that the mean dis- 
tances of the planets contain no long-period terms of orders zero or one, 
which corresponds to the theorem of the invariability of the mean distances. 
‘ «Sur les solutions périodiques et le principe de moindre action,’ @. RB, cxxiii. 
pp. 915-8 (1896) ; ‘Les solutions périodiques et le principe de moindre action,’ zbid. 
exxiv. pp. 713-6 (1897). 
2 «On the part of the Parallactic Inequalities in the Moon’s motion which is a 
function of the mean motions of the Sun and Moon, Amer. Jour. Math. xiv. pp. 
141-60 (1892); ‘The Elliptic Inequalities in the Lunar Theory,’ ibid. xv. pp. 244-63, 
321-38 (1893); ‘Investigations in the Lunar Theory,’ ibid. xvii. pp. 318-58 (1895). 
% «The Periodic Solution as a first approximation in the Lunar Theory,’ Ast. Jowr. 
Xv. pp. 137-43. 
* Transactions of the Physical Section of the Imp. Soe. of Nat. Sc. Moscow, viii. 
5 «Sur les formules générales de la Mécanique Céleste,’ Annales de la Fac. de 
Toulouse, iv. K,35 pp. 
