526 
PROFESSOR CH ALLIS ON THE PROBLEM OF THREE BODIES. 
3. The foregoing preliminaries having been gone through, the order in which the 
approximate integration is to be elfected may now be stated. As the approximation 
is to proceed according to the disturbing force, the equations (6.) and (7.) must first 
be integrated omitting the terms involving R. We shall thus obtain values of r and 
6 as functions of t and arbitrary constants, just as in the case of the problem of two 
bodies, and these constants may be designated by the letters usually employed in that 
problem. As the exact values of the functions would be unsuitable for carrying on 
the approximation, they may be expanded in series proceeding according to the 
powers of the arbitrary constant e to as many terms as we please. In like manner 
the functions which express the values of r' and & in terms of t, may be expanded 
according to the powers of e'. When these values of r, 6, r' and & have been substi- 
tuted in the right-hand side of the equation (7.), that side becomes a function of t 
and constants ; and supposing the integrations indicated to have been effected, and 
the result to be Q, we shall have 
A® 
r 
+C=2Q, 
Q being a small quantity of the order of the disturbing force. 
Hence 
— — C+2Q| 
&c. being neglected. In the second term we may substitute for r in tefms of f 
from the first approximation, which gives 
dr 
2a 7 ,/ , 2a \ u, 
dr‘ 
dr 
Supposing, therefore, that by the first approximation ^^-2 =/’(05 we obtain 
dr 
dt{\+Qf{t)) = 
0 ’ 
This equation being integrated, a relation is found between r, t and arbitrary con- 
stants, by means of which r is to be expressed in a series proceeding primarily 
according to the disturbing force, and subordinately according to the quantities e, e' 
and <y. This value of r is next to be substituted in the equation (6.), which, being 
put under the form 
hdt ( , \ dt 
shows that the right-hand side then becomes a function of t and constants, and that 
by integration ^ may be obtained in a series proceeding according to the same law of 
arrangement as the series for r. 
