220] 
519 
220. 
NOTE ON A THEOREM OF JACOBI’S, IN RELATION TO THE 
PROBLEM OF THREE BODIES. 
[From the Monthly Notices of the Royal Astronomical Society, vol. xxn. (1861), 
pp. 76—78.] 
The following theorem of Jacobi’s (Comptes Rendus, t. III., p. 61 (1836)) has not, 
1 think, found its way in an explicit form into any treatise of physical astronomy. 
The theorem is as follows, viz. “ Consider the movement of a point without mass 
round the Sun,, disturbed by a planet the orbit of which is circular. Let xyz be the 
rectangular coordinates of the disturbed body, the orbit of the disturbing planet being 
taken as the plane of xy, and the Sun as the centre of coordinates ; let a' be the 
distance of the disturbing planet, n't its longitude, m! its mass, M the mass of the 
Sun : then we have, rigorously, 
M ,( 1 
+ m l/.v, , _.o—£>..// 
X cos n't + y sin n't 
(x 2 + y 2 + z 2 f [(¿r 2 + y 2 + z 2 — 2a (x cos n't + y sin n't) + a 2 f 
This is therefore a new integral equation, which, in the problem of three bodies, 
subsists, as regards the terms independent of the eccentricity of the disturbing planet, 
and which is rigorous as regards all the powers of the mass of such planet. In the 
Lunar Theory the Earth must be substituted in the place of the Sun, and the Sun 
taken as the disturbing planet.” 
To prove the theorem, as expressed in polar coordinates, I take the equations of 
motion in the form in which I have employed them in my “ Memoir on the Theory 
of Disturbed Elliptic Motion” (Memoirs, vol. xxvii. p. 1 (1859)), [212], viz.