PROFESSOR CHALr.IS ON THE PROBLEM OF THREE BODIES. 
543 
Problem of Three Bodies. The difference of the analytical results of the two methods 
may be thus explained. The equation obtained by putting ^ = 0, viz. h^—2^r-\-Qr^ 
—ljj'r‘^—0, may be shown to have three positive roots if C be positive, so that analyti- 
cally there are three apsidal distances. The first method, by embracing the third 
apsidal distance (no step being taken to exclude it), applies to the other two only in 
an expanded form, the expansion giving rise to terms containing the factor t. The 
other, by commencing the approximation with the equation 
|^+^-^+c=o, 
restricts the application of the solution to the part of the curve which has two apsidal 
distances, and accordingly finds the function of t which in the other method is ex- 
panded. The method of the variation of parameters, by the very nature of the pro- 
cess, restricts the analysis in the Problem of Three Bodies to two apsidal distances, 
and on this account is successful in determining the motion of the apse. 
Again, I think it important to remark that the solution of the Problem of Three 
Bodies, as here proposed, applies equally to the Lunar and the Planetary Theories. 
The Problem of the Moon’s motion does not differ in the analytical treatment it 
requires, from that of the motion of a Planet. In the one case as well as the other the 
approximation ought to be conducted primarily according to the disturbing force, 
which is assumed to be small compared to the principal force, and secondarily 
according to the form of the orbit, which is assumed to differ little from a circle. It 
is not necessary to take account of the ratio of w' to n in arranging the developments, 
but only in estimating the magnitude and importance of the terms resulting from 
the integrations. The possibility of effecting the integrations is the proper proof of 
the correctness of the process, and of its being adequate to give the development which 
is alone appropriate to the question, and which must result from every process that 
is in all respects legitimate. After making any assumption respecting the analytical 
form of the solution (as in the Lunar Theory is done by introducing the constants c 
and g), there can be no certainty that the solution will not at some stage become 
empirical. Probably the reason that the process which succeeds for a planet has not 
been applied to the moon, is the difficulty of extending it to the square and higher 
powers of the disturbance (which in the Lunar Theory it is necessary to take into 
account), and of embracing in the same operation both the periodic and the secular 
inequalities. The method I have exhibited in this communication appears to meet 
this difficulty by evolving simultaneously both kinds of inequalities by a process 
which obviously may be extended to higher powers of the eccentricity and the dis- 
turbing force. Such an extension would require nothing more than great labour in 
executing the analytical details. 
