PROGRESS OF THE SOLITION OF THE PROBLEM OF THREE BODIES. 157 
the lunar parallax, in the generalised form previously given by the author, 
are shown to be simple deductions from a single equation. 
Researches relating to the convergence of the trigonometric series of 
dynamical astronomy were published in 1896 by Charlier! and in 1898 by 
Poincaré.? The former, by expanding in descending powers of m the 
coefiicient of the mth term in such a series, arrived at the conclusion that 
the convergence can be augmented by dividing the function expressed into 
two parts, one of which depends on the first terms in these expansions of 
the coefficients. In Poincaré’s paper the author first connects the series of 
the older theories, in which the time occurs explicitly, with the new 
expansions, and then observes that the slow convergence of the latter 
is to some extent compensated for by the fact that the terms can be 
grouped together in such a way that, although the individual terms of a 
group may be large, yet their sum is small. The latter part of the 
paper is devoted to showing how the expansions which represent periodic 
and asymptotic solutions can be derived from the general expansions. 
§ VII. The Impossibility of Certain Kinds of Integrals. 
Poincaré’s theorem on the non-existence of uniform integrals of the 
problem of three bodies, other than those already known, has already 
been reviewed in § VI. Before the publication of Poincaré’s memoir, 
however, an important theorem on the non-existence of algebraic integrals 
had been obtained by Bruns. 
In Bruns’s paper, the differential equations of the problem are first 
taken, in their unreduced form, as a system of the 6th order ; they can 
be written 
sees BM flan Voy ++ + L3ny rp C=), 2, sae 3n) 
where ¢ denotes the sum of all the mutual distances of the bodies ; the 
reason for introducing ¢ is that the quantities / are rational functions of 
Hy, Loy.+. Lan, ¢, Whereas they would be irrational functions of 
%\, Xy,—%3, alone. ¢ is a function of the a’s, given as a root of an 
algebraical equation. 
Bruns supposes that this system of equations possesses an integral of 
the form ¢(x,, 2%)... X3ny Yir Yo +++» Yan), Where B09 and where ¢ 
is an algebraic function of its arguments. must therefore be a root of 
an algebraic equation whose coefficients are rational functions of the 
quantities x, y, and which we may take to be irreducible. On differen- 
tiating this, it appears that either the coefficients of the algebraic equation 
in » are themselves integrals, or else » satisfies an equation of lower 
degree, whose coefficients are rational in the quantities x, y, c. In this 
way it is proved that all integrals which are algebraic functions of the 
quantities x and y are algebraic combinations of other integrals which 
are themselves rational functions of the quantities 2, y, and ¢. We need 
1 «Ueber die trigonometrischen Entwickelungen in der Stérungstheorie,’ Asé, 
Nach. cxli. pp. 273-8. 
? «Sur la fagon de grouper les termes des séries trigonométriques qu’on rencontre 
en mécanique céleste,’ Bull. Astv, xv. p. 289-310. 
% ‘Ueber die Integrale des Vielkérper-Problems,’ Berichte der Kgl. Stéichsischen 
Ges. der Wiss. zu Leipzig, 1887, pp. 1-39, 55-82; Acta Math. xi. pp. 25-96. 
