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method of osculating variation, before briefly described ; from which 

 are deduced the formulae for the osculating variations of elliptic ele- 

 ments. This method is capable of being applied to the planetary 

 and lunar theories, as well as that of tangential variation ; but the 

 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 a (a-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)*.+ (31)*, + (12)*,. 

 (23) +(31) +12 



with similar expressions for Y and Z; (12) being r 12 + m m. 2 x^ 12 , 

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

 w 2 ; their ratios being denoted by the proportion 



2 r o 



: *! : : 



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

 and it is shown that 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 

 the 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 



