532 
PROFESSOR OH ALLIS ON THE PROBLEM OF THREE BODIES. 
Substituting- Q for the sum of the terms on the right-hand side of this equation, and 
neglecting &c., vve have 
rdr Qdt / . . \ 
~ V' 2/xr — Cr^~ sin^^j ^ ^ ” eCOS/>), 
because by the first approximation 
2]a 
r 
dt dt 
dr'^ od-e^rd- ( I + 4e cos^) 
dt"^ 
Consequently 
nt-\-z — nT = COS' 
a — r 
ae 
2 ( 1 — 4e cos p), 
( 8 .) 
the constants a, e, i and ts- having the same signification as heretofore*. 
10. Before proceeding to effect the integration above indicated, it will be right to 
remove certain analytical difficulties presented by the form of the equation. First, it 
may be urged that as Q contains the first power of e, the coefficient of the last term 
might become infinite if e were indefinitely small, and the equation would no longer 
hold good. But it has already been proved (art. 5.) that e and the disturbing force 
vanish together, from which it follows that the quantity, jX disturbing force, may 
approach a finite value or zero as e diminishes. Again, it will be seen, if the quantity 
to be integrated be put under the form that the factor x,(0 becomes infinite each 
time sin/?=0, and that the development fails for the values of t that satisfy this equa- 
tion. But it is well known that an analytical circumstance of this kind will not prevent 
our obtaining in the final analysis the correct development ofr, provided the integration 
above indicated can be effected, and that the failure must admit of some interpretation 
relative to the proposed problem. Nov/ it is not difficult to point out the significance 
(l“~T 
of the failure in this instance. Let us suppose that cos“‘ Then the are (p 
can differ from only by a stnall quantity, and we have exactly r=a{l—e cos (p). 
Hence as the quantities a and e are absolutely constant, it would seem that the 
maximum and minimum values of r are «(l+e) and a(l—e) in every revolution of 
the disturbed body. This inference is manifestly untrue, and the reason that it has 
not been legitimately deduced is, that the above-mentioned failure occurs when r 
approaches a maximum or minimum value. The failure has, therefore, an important 
bearing on the problem, as showing that the maximum and minimum values of the 
radius-vector are not of constant magnitude. I proceed now with the integration. 
* See Note (B) at the end of the paper. 
