DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
:U5 
This is easily obtained from (109.) and (110.); and in like manner 
tani=— 
2 O’ +Pi +p cos I 
I shall now recapitulate the results of the last supposition, so as to exhibit in one 
view the transformed differential equations of the problem of three bodies. It will 
be as well to repeat also the explanation of the symbols. 
92. Signification of the Symbols. 
M, m, are the masses of the sun, and of the two planets, and 
a and e are the semiaxis and excentricity of the instantaneous ellipse described by 
m about the sun. 
and e, have the same meanings with reference to 
I is the inclination of the plane of the former ellipse to that of the latter. 
01 are the longitudes of the two planets, measured in the planes of their orbits 
Jrom the common line of nodes. 
r, r, their radii vectores. 
tzr, d, the longitudes of the perihelia, measured likewise in the planes of the orbits 
from the line of nodes. 
s, two elements such that ^ ndt-\-s, ^nft-{-Si are the mean longitudes, where n, n^ 
are defined as usual by the equations 
T\\en^ndt-\-z — d is the mean anomaly of m, and r, 6 are functions of the mean 
anomaly and mean longitude given by the laws of elliptic motion. The same is true 
for m^, mutatis mutandis. 
X is the angle between the radii vectores, so that ' 
cos %= cos 0 cos sin 0 sin 0^ cos I. 
Let § be the distance between the planets, so that 
— 2rr^ cos %. 
Q, are the disturbing functions, defined by the equations 
Q=mm, 
r cos 
X 
AS 
^ /I 7*. COSVN 
and, when expressed in terms of the elements and t, are functions of 
a, a^, e, e^, I, 1 ndt-\-z, ? ndt-^z—i;^, I nfit-\-z^, I nft-\-i^ — 
Jo Jo Jo Jo 
When Q, are considered on the one hand as expressed in this way, and on the 
other, in their original form as functions of r, 0, 0^, and I, we have, as applied to 
either of them, 
d d d d d d 
dS ds'dzj"' dSi dsid'sj^ 
