472 Royal Society. 



advantages of this method did not appear to be such as to justify 

 the actual expansion of the formulae for these theories. The author, 

 however, shows that with reference to any system of three bodies, 

 the equations of motion for each body naturally assume the form 



x" + n-{x—X)=0, &c. 

 (being the system solved by this method) ; and that the X, Y, and 

 Z are absolutely the same for each of the three bodies. This is 

 shown by demonstrating, that at any given moment the three lines 

 which represent the direction of the force acting on each of the 

 three bodies all pass through the same point, which is denominated 

 the centre of force. The coordinates of this common centre of force 

 are, 



(23),r, +(3lK+(12>r3 . 

 ^- (23) +(31) +12 ' 

 with similar expressions for Y and Z; (12) being i\^-^mm.^-)(ry,, 

 )^ denoting the law of force, &c. Each body has its own value of 

 »" ; their ratios being denoted by the proportion 



fl* « J. 



2 , 2 . 2 , 2.*? , * 31 ^ 11 



«X • «2 • '^3 • • ^,.^^ • ^^^^ • ^r^^- 



The invariable plane of this system of three bodies is then found ; 

 and it is shown tbat the nodes of the three orbits upon this plane 

 are always in a certain relative position, constituting a kind of tri- 

 angle of equilibrium about the centre of force ; resulting, in the 

 limiting case where one of the three bodies is infinitely larger than 

 tlie other two (or in what is denominated the undisturbed Problem 

 of Three Bodies), in an exact opposition of the two nodes of the 

 orbits of the latter two bodies upon the invariable plane of the 

 system. 



The formulae for the osculating variation of elements are then 

 applied to a system of three bodies, of which one possesses a pre- 

 dominating magnitude, so far as is necessary to determine the move- 

 ment of the planes of the orbits ; and it is readily shown that, if we 

 consider only the first order of the disturbing force, the inclination 

 of the plane of each orbit to the invariable plane is absolutely con- 

 stant; and that the two nodes are always in opposition to each 

 other, and move with a uniform angular velocity round the inva- 

 riable plane. 



This theorem is then extended to a system of n bodies moving 

 about a central predominant body ; and it is shown that the ag- 

 gregate effect of the disturl)ing forces of such a system upon the 

 plane of any one of the bodies can always be represented by stating 

 that its node upon a certain fixed plane revolves with a uniform 

 angular velocity, the plane of the orbit always remaining at the 

 same inclination to the fixed plane. The rate of this angular move- 

 ment, and the coordinates of the fixed plane upon which the move- 

 ment takes place, are found by means of formulae of remarkable 

 simplicity. These three quantities may be ascertained once for all 

 for each planet (viz. the inclination of the fixed plane on which the 



