528 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
the elliptic theory, let g—cr be put forwT. Then substituting jo for — nr, we have, 
as is known, the following expansions of r and d to the third power of e: 
r 3e^ 
-= 1 — e COS /?+ 2 (1 — COS 2jw)H--^(cosjb — COS 3p) 
5g2 ^ g3 ^23 \ 
^=g+w/+2esinjo + — sin 2p+4( ^ 3jo — sin/? j. 
These values of r and & (excluding the terms involving for the sake of avoiding 
long calculations) will be employed in proceeding to the second approximation. It 
is evident that C, h, tu and g must be regarded as the arbitrary constants of the inte- 
gration however far the approximation be carried, no other arbitrary quantities being 
introduced by the process. The quantities a and e are given functions of C and A, 
and at present they have no other signification. 
5. Before proceeding further, an inference may be drawn from the equation (7.) 
which will be useful hereafter. When the foregoing values of r and 6 are substituted 
in the right-hand side of that equation, the constant e will be a multiplier of that 
side, independently of any limitation of the orbit of w'. Now let, if possible, e=0. 
h^C 
Then 1=—, and the equation (7.) becomes 
dr'^ 
Since the relation [//=h^C shows that C must be positive, it follows from the above 
equation that ^=0 and ^=C, or that the orbit of w is a circle whose radius is equal 
to But the orbit of m cannot be exactly a circle independently of the form and 
magnitude of the orbit of m', unless the disturbing force be indefinitely small. Con- 
sequently the supposition that e=0 draws with it the inference that the disturbing 
force vanishes. At the same time, the supposition that the disturbing force vanishes 
must leave e an arbitrary quantity, because on this supposition the problem is that of 
elliptic motion, and e is the eccentricity of the orbit. These conditions may be ana- 
lytically expressed by such an equation as e^=el-\-km', k being positive and of fixed 
value, and being arbitrary*. 
Second Approximation. 
6. The first step towards expressing the right-hand side of the equation (7.) as a 
function of t, is to expand the quantity R in a series proceeding according to cosines 
ot multiples of the arc 0 — Let 
R=Rq-{-R) cos (^— ^')-1-R2Cos 2(^— ^')-|- &c. 
= Ro-l-2 . R^cos s{6—d), 
the values of .v being the integers 1, 2, 3, &c. Also let r=a(l+M), 'r'= a' (1 -[-?/), 
* See Note (A) at the end of the paper. 
