544 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
NOTE (A). 
Received July 7, 1856. 
It has been shown in art. 5 from a 'priori considerations, that if the constant e=0, 
the disturbing force vanishes, and if the disturbing force vanishes, e is arbitrary. 
Hence it appears from the development of the radius-vector in art. 16, that the eccen- 
tricity of the disturbed orbit and the disturbing force are related in such a manner 
that if the eccentricity =0, the disturbing force vanishes, and if the disturbing 
force =0, the eccentricity remains arbitrary. The particular relation which satisfies 
these conditions ought plainly to result from the solution of the Problem of Three 
Bodies, and it may, therefore, be worth while to inquire how far such a result can 
be deduced from the integrations effected in the foregoing approximate solution. 
Now the expressions for the variations of the eccentricity and of the longitude of 
the apse obtained in art. 17, are identical with those given by the method of the 
variation of parameters. Hence for the present purpose I may make use of the 
deductions from those expressions which are usually given in treatises on the Plane- 
tary Theory. Referring to Pratt’s ‘ Mechanical Philosophy,’ art. 385, we have the 
equations, 
D^=: — (2BD-t-CD') . . . (1.) E/i=~(2BE-4-CE') . . . . (2.) 
j/, fj, 
D'g=— (2B'D'-fCD) . . . (3.) E'A=— (2B'E'-hCE) . . . (4.) 
ft ft 
e7=D^-l-E®-l-2DE cos /}• (5.) 
6''®=D'^-1-E'^-1-2D'E' cos {(g-— /} (6.) 
g or A = 
nam'B + n'a'mB' 
{ {nam'B — n'a’tnB'y-^nn'aa'mm'C^}^ 
( 7 .) 
In these equations B, B' and C are known quantities independent of the eccentricities 
and longitudes of the apses, and e are respectively the eccentricities of the orbits 
of the disturbed and disturbing bodies, k and / are arbitrary quantities, and D, D', E, E' 
are also arbitrary, excepting so far as they are connected by the first four equations. 
Let, if possible, e^ = 0 independently of the time. Then it follows from (5.) that D = 0 
and E=0. Hence, since e[ does not consequently vanish, it appears by (6.) that D' 
and E' do not on this supposition both vanish, and, therefore, by the first or second 
equation, that m'=0. Again, let m!=0. Then by (1.) and (2.), Dg=0 and E/i = 0, 
and by (7.), one of the quantities g and h vanishes. Hence one of the quantities D 
and E vanishes and the other remains arbitrary. Hence also is arbitrary. These 
results confirm the reasoning in art. 5. 
