542 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
fact extend beyond the first power of the disturbing force, so far as it relates to these 
two elements. 
If the approximation be made to include generally the square of the disturbing 
force, and the values of r and & in art. 16, and the like values of r' and be used 
for that purpose, terms may arise containing coefficients which have f for a factor. 
These terms may be converted into periodic functions of the time by the application 
of the principles exhibited above, but in that case the differential equations by which 
E, E', n and 11' are found will be of the second order, and the periodic functions 
will be more completely determined. 
The inferences (1), (2), (3) and (4) respecting the secular variations of the elements, 
although obtained in a manner quite new, agree exactly with those deduced from 
previous solutions of the same problem. 
18. Having now obtained the developments of r, r' and inclusive of both 
periodic and secular inequalities, to an extent which is sufficient for most of the appli- 
cations of the Planetary Theory, I shall reserve for a future opportunity the investi- 
gation of the inequalities in latitude, and shall then take occasion to show in detail 
how this method adapts itself to the determination of the motions of the moon. At 
present I propose, in concluding this memoir, to make a few general remarks on the 
Problem of Three Bodies. 
It has been already observed, that the solution here adopted introduces no terms 
containing ent in the coeflScients. These terms are to be distinguished from those 
whose coefficients contain which, as we have seen, have reference to secular 
variations of the eccentricity and of the motion of the apse, and would vanish with 
the eccentricity of the orbit of the disturbing body. The former relate to the motion 
itself of the apse, and are not peculiar to the Problem of Three Bodies, occurring in 
fact in cases where the force is directed to a fixed centre. To illustrate this remark. 
let us suppose the force directed to a fixed centre to be Then, the differen- 
tial equation for finding the orbit being 
d^‘- 
r 
1 [X, ixJr^ 
r 
: 0 , 
let this equation be integrated by successive approximations, first neglecting the last 
term, and then substituting in that term the value of r given by the first approxima- 
tion. By this process a term containing t in the coefficient will be introduced, and 
the motion of the apse will fail of being ascertained. But if, instead of this process, 
the equation dr’^ ‘lix , , , ^ 
^^2+^2- ^ -PC— 0 
be obtained, and its approximate integration be conducted according to the powers 
of (jJ, no such term will arise, and the motion of the apse will be determined. The 
latter process is exactly analogous to steps employed in the foregoing solution of the 
