152 REPORT—1899. 
Poincaré’s method for the direct calculation of periodic solutions of 
dynamical systems was modified in 1895 by Kobb,! so as to be applicable 
to the problem of three bodies. 
Darwin in 1896 published a preliminary EoGie? of a paper ® which 
appeared in 1897, and in which a large number of periodic solutions are 
calculated numerically. The case considered is the restricted problem of 
three bodies ; two of the bodies, S and J, revolve round each other in 
circular orbits, and the mass of the third body P is zero. Darwin finds 
six families of periodic orbits ; in one (Planet A), P describes a closed 
path round §, as in Hill’s periodic orbit ; in two others (oscillating Satel- 
lites a and 6) P oscillates about a position on the line joining § and J, as 
in the libratory motions of Burrau, Thiele, and Perchot and Mascart ; and 
in the remaining three (Satellites A, B, C), P describes a closed path 
round J. In the numerical work, the mass of S is supposed to be ten 
times the mass of J. When different values are assumed for the constant 
of energy, the orbits of these families change their form, pass from 
stability to instability and vice versd, and even go out of existence 
altegether. 
Another class of periodic solutions of the restricted problem of three 
bodies was found in 1898 by Schwarzschild. 
An application of Poincaré’s theories in a different direction was made 
in 1893 by Perchot.® In the first part of his memoir the coefficients of 
the principal periodic inequalities of the longitudes of the lunar node and 
perigee are calculated ; the author takes the equations in Delaunay’s 
form, and applies the theory of periodic solutions. In the second part, 
the secular variations of the eccentricities and inclinations are discussed, 
with the aid of Poincaré’s theory of stability. 
The theories of periodic solutions, characteristic exponents, asymptotic 
solutions, and the non-existence of uniform integrals were somewhat 
more completely discussed in 1892 by Poincaré® himself in the first 
volume of his treatise on the new developments of dynamical astronomy. 
The second volume, which was published in 1893, and contains a good 
deal of matter which had not appeared in the memoir of 1891, opens with 
a chapter on asymptotic expansions ; after this the author discusses, by 
Jacobi’s method, Lindstedt’s theory of the solution of the equations of 
dynamics by means of sums of periodic terms, using his own proof of its 
applicability, as given originally in C. &. cviii. Newcomb’s method is 
shown to be fundamentally equivalent to Lindstedt’s. Lindstedt’s theory 
is then applied, firstly, to prove a result obtained by Poincaré? in ‘C. R.’ 
exiv. regarding the expression of the secular inequalities in the planetary 
1 «Sur le calcul direct des solutions périodiques dans le probléme des trois corps,’ 
Ofversigt at K. Sv. Vet.-ak. Forh. lii. pp. 215-22. 
2 «On Periodic Orbits, Brit. Assoc. Report, 1896, pp. 708-9. 
3 «Periodic Orbits, Acta Math. xxi. pp. 99-242; Math. Annalen, li. pp. 523-83. 
+ « Ueber eine Classe pericdischer Losungen des Dreikérperproblems,’ Ast. Mach. 
exlvii. Pp. 17-24; ‘Ueber weitere Classe periodischer Losungen des Dreikérper- 
problems,’ idid. pp. 289-98. 
5 «Sur les mouvements des eae et du périgée de la lune, et sur les variations 
séculaires des excentricités et des inclinaisons, Annales de le. Norm. Sup. (5) 
x. suppl. pp. 3-94. 
6 Les Méthodes Nouvelles de la Mécanique Céleste, Paris, Gauthier-Villars, vol. i. 
1882, vol. ii. 1893, vol. iii. 1898. 
7 «Sur lapplication de la méthode de M. Lindstedt au probléme des trois corps,’ 
C. R exiv pp, 1305-9. 
