PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 153 
theory as sums of periodic terms, and, secondly, to effect the general 
expansions in the problem of three bodies; as explained in the paper 
referred to, there is a difficulty here, since in Kepler’s ellipse the node 
and perihelion are fixed, and thus there is a linear relation between the 
mean motions of the arguments used. This difficulty is surmounted, and 
another is considered in the following chapter, arising from the fact that 
if the eccentricities are very small (supposing that ¢ is used as one of the 
variables, and not e cos a and ¢ sin a), some of the developments break 
down. It isshown that this difficulty can be avoided by taking as starting- 
point a periodic solution instead of Kepler’s ellipse. 
The author then proceeds to discuss the convergency of Lindstedt’s 
expansions ; his results in this connection were disputed by Hill,’ and led 
to some controversy. 
After some interesting remarks on the theorem of the secular invaria- 
bility of the mean distances, Poincaré proceeds to show how the coefficients 
in Lindstedt’s series can be calculated directly, without the complicated 
transformations which were introduced in the proof of their existence ; 
and then a new way of forming Lindstedt’s series is explained, in which 
half of the original equations of motion are replaced by the equation of 
energy and certain equations involving an auxiliary function S. Two 
equalities which can be used in the verification of these processes were 
given in 1895.? 
The first half of the book may be said to centre round a theorem, 
which may be stated as follows :— 
Let the equations of dynamical astronomy be given in the form 
(1) dx,_@F dy,_ oF 
ey ee 
GEM Oy, aR ORE da Painaacdin 
The function F is periodic in the quantities y, and may depend on the 2’s 
in any manner. Moreover, certain of the terms are small in comparison 
with others, and the order of magnitude of the different terms may be put 
in evidence by introducing a small quantity », and developing F in 
ascending powers of j, in the form 
F=F)+ypF,+p2F, +... ; 
F,, does not involve the quantities y. 
Then it is proved that the equations (1) can be formally satisfied by 
series of the form 
9 fo =a Pt poO+woPt... Sanaed 
@) Lyi Hey Fey P +... CaL ay? 
where the quantities ~{° and y,” are periodic functions of the quantities 
wW=nita,; (i=1,2,...) F 
) Hill (1895), ‘On the Convergence of the Series used in the subject of Perturba- 
tions, Bull. Amer. Math. Soc. ii. pp. 93-7. Poincaré (1896), ‘Sur la divergence des 
séries de la Mécanique Céleste,’ C.2. cxxii. pp. 497-9. Hill (1896), ‘ Remarks on the 
Progress of Celestial Mechanics since the middle of the century,’ Bull. Amer. Math. 
Soc. ii. pp. 125-36. Poincaré (1896), ‘Sur la divergence des séries trigonométriques, 
C. R. exxii, pp. 557-9. 
* «Sur un procédé de vérification, applicable au calcul des séries de la Mécanique 
Céleste,’ C. 2. exx. pp. 57-9, : 
