194, REPORT—1862. 
the “Mémoire sur la Mécanique Céleste,”’ read at Turin in 1831, but it is 
reproduced in the memoir * Considérations nouvelles sur les suites &c.,” Mem. 
d’Anal, et de Phys, Math. t. i. (1840); and see also the memoirs in ‘ Liou- 
ville’s Journal’ by Puiseux, and his Note i. to vol. ii. of the 3rd ed. of the 
‘Mécanique Analytique’ (1855), There are on this subject, and on subjects 
connected with it, several papers by Cauchy in the ‘Comptes Rendus,’ 1840 
_et seq., which need not be particularly referred to. 
The Problem of two Centres, Article Nos. 41 to 64, 
41. The original problem is that of the motion of a body acted upon by 
forces tending to two centres, and varying inversely as the squares of the 
distances ; but, as will be noticed, the solutions apply with but little variation 
to more general laws of force. 
42, It may be convenient to notice that the coordinates made use of (in 
the several solutions) for determining the position of the body, are either the 
sum and difference of the two radius vectors, or else quantities which are 
respectively functions of the sum and the difference of these radius vectors*. 
If the plane of the motion is not given, then there is a third coordinate, 
which is the inclination of the plane through the body and the two centres 
to a fixed plane through the two centres, or say the azimuth of the axial 
plane, or simply the azimuth. 
43, Calling the first-mentioned two coordinates r and s, and the azimuth yp, 
the solution of the problem leads ultimately to equations of the form 
dr _ ds _ Adr, pds pdr | ods 
VE WS “VRS “HVE WS 
where R and § are rational and integral functions (of the third or fourth 
degree, in the case of forces varying as (dist.)—®) of 7, s respectively (but 
they are not in general the same functions of r,s respectively); \ and p are 
simple rational functions of 7, and » and o simple rational functions of s; so 
that the equations give by quadratures, the first of them the curve described 
in the axial plane, the second the position of the body in this curve at a given 
time, and the third of them the position of the axial plane. In the ordinary 
case, where R and § are each of them of the third or the fourth order, the 
quadratures depend on elliptic integralst ; but on account of the presence in 
the formule of the two distinct radicals /R, /§, it would appear that the 
solution is not susceptible of an ulterior development by means of elliptic and 
Jacobian functionst similar to those obtained in the problems of Rotation and 
the Spherical Pendulum. 
44, It has just been noticed that when R, S are each of them of the fourth 
order, the quadratures depend on elliptic integrals; in the particular cases 
mdr __ nds 
VR VS 
* Tf v, wu are the distances of the body P from the centres A and B, @ the distance AB, 
é, the angles at A and B respectively, and p=tan } % tan 4, gq=tan 3 +tan } y, then, 
in which the relation between 7, s is of the form >» Rand § being 
as may be shown without difficulty, v+u=a ioe o—uaazyt so that p and q are 
a a ofv-+u and »—w respectively ; these quantities p and q are Euler’s original coordi- 
nates. 
+ The elliptic integrals are Legendre’s functions F, B, 1; the elliptic and Jacobian 
functions are sinam., cosam., Aam., and the higher transcendants 0, H. 
