PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 151 
shown that these equations must be satisfied identically ; thus ¢ is a 
function of F, and so the two integrals » and F are not distinct. 
Hence the result. The system possesses no integral (other than the 
integral F), which is a uniform analytic function Of 21, Loy Yi. Yo: » for 
all values of y, and yo, for small values of », and for values of w, and x, 
contained within certain limits, and which is periodic in y, and 7. 
This forms an important complement to Bruns’s result, which will be 
reviewed in the next section of this report. 
The last section of Poincaré’s memoir is occupied with the extension 
of the results, which have been obtained for systems with two degrees of 
freedom, to more general systems, i.e. to the problem of m bodies. 
Poincaré’s paper gave a fresh stimulus to the investigation of periodic 
solutions. In 1890 v. Haerdtl! calculated numerically two cases of the 
restricted problem of three bodies. Charlier? in 1892 discussed the same 
cases by means of expansions proceeding in ascending powers of the time, 
and the same author? in 1893 found a set of periodic solutions of the 
problem of three bodies in a plane, whose expansion involves four arbi- 
trary constants ; the mutual distances of the bodies are given as series of 
ascending powers of a quantity 
{a? + a? — 2a’ cos (dt + «)}, 
the coefficients in the series being constants. 
Callandreau‘ in 1891 discussed the equations which lead by successive 
approximations to solutions differing but little from periodic solutions. 
Lord Kelvin * in 1892 traced by graphic methods a looped orbit, which 
may be regarded as a continuation of the set of periodic solutions which 
Hill believed to be terminated by the moon of maximum lunation. 
Coculesco ® in 1892 proved the stability (in both Hill’s and Poisson’s 
senses) of the motion in one of the cases treated by v. Haerdtl. The 
motion of the same system, under fresh conditions of projection, was 
investigated in 1894 by Burrau? ; in the second paper he considers those 
purely libratory motions in which the zero particle does not, in the rela- 
tive movement, circulate round either of the other bodies, and finds that 
the series of solutions is terminated by an orbit of ejection, in which the 
zero particle starts from a collision with one of the other bodies. These 
libratory orbits were further discussed in 1895 by Thiele,’ and (by use of 
Poincaré’s theory) by Perchot ? and Mascart. 
1 ©Skizzen zu einen speciellen Fall des Problems der drei Kérper, Abhand. der 
K. Bayer. Ak. in Miinehen, xvii. pp. 589-644. 
2 «Studier dfver tre-kroppar-problemet,’ Bihang till K. Sv. Vet.-ak. Handlingar, 
xviii. No. 6. 
3 Thid. xix. No. 2. 
4 «Sur quelques applications des théories concernant les solutions particulicres 
périodiques du probléme des trois corps et l'intégration des équations différentielles 
linéaires 4 coefficients périodiques, Bull. Astr. viil. pp. 49-67. 
5 «On Graphic Solution of Dynamical Problems, Brit. Assoc. Report, 1892, 
pp. 648-52; Phil. Mag. xxxiv. p. 447. 
° ‘Sur la stabilité du mouvement dans un cas particulier du probléme des trois 
corps,’ C. #. cxiv. pp. 1339-41. 
7 Recherches numériques concernant des solutions périodiques d’un cas spécial 
du probléme des trois corps,’ A. WV. cxxxy. pp. 233-40, cxxxvi. pp. 161-74. 
8 Tbhid. cxxxviii. pp. 1-10. 
® «Sur une classe de solutions périodiques dans un cas particulier du probléme 
des trois corps,’ C. 2. cxx. pp. 906-9; Bull. Ast. xii, pp. 329-52. 
