PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 121 
Report on the Progress of the Solution of the Problem of Three Bodies. 
By HE. 'T. WHITTAKER. 
Introduction. 
Tur present Report is the fulfilment of the author’s engagement to draw 
up a report on the planetary theory for the Association. The above title 
has been adopted in place of that originally chosen, as indicating more 
definitely the aim of the Report. 
The fundamental problem of dynamical astronomy is that of deter- 
mining the motion in space of any number of particles which attract each 
other according to the Newtonian law. ‘The solution of the problem 
depends on the integration of a system of differential equations ; and 
various methods have been given for the solution of the equations by 
means of infinite series of known functions. The methods are, however, 
in general cumbrous ; the convergence of the series employed has only 
recently been considered with any success, and the true nature of the 
integrals of the problem is unknown. 
The theory has hitherto been developed chiefly with the object of 
determining the motion of the moon and planets. While, however, the 
lunar and planetary theories are, both of them, attempts to solve the 
problem of three bodies, yet the results of the two theories are quite 
different in form ; this is owing to the fact that the assumptions on which 
the approximations are based are not the same in the two cases. Thus 
it is known that if the masses of all but one of the particles are zero (ve. 
do not exert any attraction on each other), these particles will circulate 
round the remaining particle in elliptic paths ; and soa method of approxi- 
mation, known as the planetary theory, has been developed, in which it is 
supposed that the mass of one body preponderates and the other bodies circle 
round it. In the lunar theory, on the other hand, it is assumed that two of 
the bodies circle round each other, while circling together round a prepon- 
derating third body. This gives rise to a solution of the problem by means 
of a different set of infinite series. 
Of course, the planetary and lunar theories do not by any means 
exhaust the list of possible methods of approximation. For instance, it is 
known that a particular solution of the problem of three bodies exists, in 
which the three particles are always at the vertices of a moving equilateral 
triangle ; and that, under certain conditions, this is a stable form of 
motion. It would therefore be possible to form a theory, analogous to 
the lunar and planetary theories, in which the approximation would be 
based on the supposition that the motion differed but little from this 
type ; and the only reason why this theory has not been developed is, that 
it is not called for by the practical needs of computers of the solar system. 
The results of the planetary and lunar theories may be regarded as 
furnishing solutions of the fundamental problem by means of infinite 
series, valid in each case only so long as the initial conditions are subject 
to certain inequalities. In addition to this, there is a considerable litera- 
ture dealing with the differential equations of the problem and their 
transformations ; and in recent years discoveries have been made relating 
to the nature and general properties of the solution, e.g. Bruns’s theorem 
that no algebraic integrals of the problem of several attracting bodies 
