226 CELESTIAL MECHANICS. I. V. 22. 



the difference would at last vanish, and the two expres- 

 sions would be equal, as was to be proved, though the 

 transformation is by no means obvioas without a demon- 

 stration.] 



For similar reasons we have 

 c'2^= W. (^'-x)(dz'-d.)-(^-.)(d^-dx) ^^^ 



d^ 



d^ 

 It is obvious that the sum, thus ascertained, will be liable 

 to the same conditions of becoming a maximum and va^ 

 nishing, which have been demonstrated respecting the radii 

 drawn to the common centre of gravity : and that the same 

 mean plane of revolution, determined from it, will always 

 remain parallel to itself. 



333. Theorem. The sum of the squares 

 of the velocities of any system of bodies, 

 taken in pairs, of which the one is considered 

 as moving round the other at rest, and mul- 

 tipUed by the products of the masses of the 

 respective pairs, is expressed by a constant 

 quantity lessened by twice the product of the 

 sum of all the masses into the sum of the reci- 

 procal forces between each pair, combined 

 with the spaces through which they act, and 

 multiplied by the products of the respective 



, (dor'— dx)2 -f (d?/— dy)3 + (dz'-^dzf 



masses ; or Xmm ^ — -^^^ ^' — r- - 



at* 

 = 0--2^mifmmFdf 



