122 REPORT—1899. 
exist, beyond the integrals of energy and momentum. In this Report it is 
intended to review the state of these various branches of the theory at the 
present time, solely in so far as they help in the mathematical discussion 
of the fundamental problem ; no attempt is made to consider numerical 
applications, or the suitability of the various developments for purposes of 
computation. On this account, many papers which are of the highest 
importance in the practical lunar and planetary theory are left unnoticed ; 
this is in some respects to be regretted, but it has been rendered necessary 
by limitations of space and time. 
The Report attempts to trace the development of the subject in the 
last thirty years, 1868-98; this period opens with the time when the 
last volume of Delaunay’s ‘ Lunar Theory’ was newly published ; it closes 
with the issue of the last volume of Poincaré’s ‘ New Methods in Celestial 
Mechanics.’ Between the two books lies the development of the new 
dynamical astronomy. 
The work will be distributed under the following seven headings ;— 
$ I.—The differential equations of the problem. 
§ II.—Certain particular solutions of simple character. 
§ III.—Memoirs of 1868-89 on general and particular solutions of 
the differential equations, and their expression by means of 
infinite series (excluding Gyldén’s theory). 
$TV.—Memoirs of 1868-89 on the absence of terms of certain classes 
from the infinite series which represent the solution. 
§ V.—Gyldén’s theory of absolute orbits. 
§ VI.—Progress in 1890-98 of the theories of $$ III. and IV 
§ VII.—The impossibility of certain kinds of integrals, 
§ I. The Differential Equations of the Problem. 
Taking any fixed axes of reference, the motion of three mutually 
attracting bodies is determined by nine ordinary differential equations, 
each of the second order, or, as it is generally expressed, by a system of 
the eighteenth order. The known fact that the centre of gravity may be 
regarded as at rest is equivalent to six integrals of the system, and so the 
system can be reduced to the twelfth order. The further fact that the 
components of angular momentum about the axes are constant yields 
three more integrals, and the system can thus be reduced to the ninth 
order. The integral of energy makes possible a reduction to the eighth 
order ; and since the time ¢ only enters by means of its differential dé, it 
can be eliminated, and the system reduced to the seventh order. A 
further simplification can be made, which was first pointed out explicitly 
by Jacobi,! though it is really contained in the work of Lagrange,” namely, 
that the variables can be so chosen that one of them © enters only by 
means of its differential dQ ; it can therefore be eliminated (and after- 
wards found by a simple quadrature), and the system can be reduced to 
the sixth order. 
Later writers have not succeeded in reducing the problem to a lower 
order than the sixth. It will be seen, however, that distinct advances 
have been made in the formulation of the equations and the theory of their 
1 ¢Sur lélimination des nceuds dans le probleme Ges trois corps,’ Credle, xxvi. 
pp. 115-31, 1843. oe 
2 «Essai sur le probléme des trois corps,’ Prix de ? Académie de Paris, ix. 1772, 
