154 REPORT—1899,. 
the quantities a; are constants of integration ; the quantities n; are 
constants (called the mean motions) which can be developed as power- 
series in p. The quantities x,“ and y;, can themselves be developed in 
series of the form 
(3) x (or y{°)=ZA cos (mw, +MyWo+ ... +m,w,+h). 
Suppose for simplicity that n=2. If the ratio of the mean motions 
is commensurable, one of the terms of the series (3) becomes infinite ; 
leaving this case, it is shown (pp. 96, 97) that incommensurable values of 
the ratio of the mean motions can be sorted into two categories—those - 
for which the series converges and those for which the series diverges— 
and in every interval, however small, there are values of the first category, 
and also values of the second category ; in particular, the series converges 
for incommensurable values whose square is commensurable. The con- 
vergence is in no case uniform. By an artifice, however, the series (3) 
can be regarded as composed of only a finite number of terms, and so the 
series (2) can be formed. 
What may be regarded as the second half of the book begins, in the 
sixteenth chapter, with an introduction to Gyldén’s theory ; the Gyldén- 
Lindstedt equation is treated by the methods of chapter ix. and by those 
of Gyldén, Bruns, Hill, and Lindstedt ; and then the author proceeds to 
the more difficult of Gyldén’s differential equations. The last three 
chapters of the volume are devoted to an exposition of Bohlin’s method, 
and to an extension in which some of the limitations of Bohlin’s work 
are removed. 
The theorem regarding the expression of the coordinates as trigono- 
metric series was still further improved! by Poincaré in 1897. It is 
shown that the coordinates in the problem of three bodies can be expressed 
by series proceeding in ascending powers of a small parameter , of the 
order of the two small masses, and of several constants p of the order of 
the eccentricities and inclinations, These series are periodic functions of 
five arguments : 
W,=Nyt+Q), Wye=Nttay wy, =Mb+e, Wo=Mobteo, wWz=1;6 + & 
Here 7, ao, &, &, €, are constants of integration ; 7, %2, 4) 1’) v3; are 
functions of «, the quantities p, and two other constants 2, and 2, and 
can be expanded as power-series in p» and the quantities p* ; the coefh- 
cients in the series depend on z, and z,. The quantities n, and , may be 
called the mean motions of the planets ; z,?and z,” may be called their mean 
distances ; a, and a, correspond to their longitudes at the epoch, the 
quantities p to the eccentricities and inclinations, the quantities w to the 
longitudes of the perihelia and nodes. The development of m, and », in 
powers of » commences with terms of order 0, while the development of 
V1, Vx) Ys Commences with terms of order one, so the w’s vary quickly and 
the w’s vary slowly. Terms whose arguments are compounded of the w’s 
only may, by analogy with the older theories, be called secular terms. 
Poisson’s theorem on the invariability of the mean distances, in its new 
interpretation, is proved in the course of the paper. The mutual distances 
of the bodies depend only on the differences of the above five arguments. 
The third (and last) volume of Poincaré’s book was published in 
1 ‘Sur Pintégration des équations du probléme de trois corps’, Bull. Astr. xiv. 
. pp. 261-70. 
