224 CELESTIAL MECHANICS. J. V. 22. 



(dX + dj^) (dA^' +dx'). . . must be equal to the sums of 

 the squares of dX, dX', . . . , da-^, dx\, . . . , since (dX + 

 da7,)2=dX2 + da7/ + 2dXdx^, and SwidXdx^zzdXSwid.r^zr 

 0, since S»?.r^=:0, and d{Xmx^)z=.0']. It appears, therefore, 

 that the quantities, concerned in the impetus and the areas, 

 are composed, first, of the quantities which would have ex- 

 isted, if all the bodies of the system had been united in their 

 common centre of gravity; and secondly, of the similar quan- 

 tities derived from the centre of gravity, considered as im- 

 moveable: and the first system of quantities being- constant, 

 it is easily understood that the second must be also constant. 

 It follows, therefore, that if we fix the origin of the co- 

 ordinates X, y, Zy x\ , . . of the equations (Z) (323) at the 

 centre of gravity, the conclusions derived from them will 

 still hold good, and the angles; of the planes concerned 

 will remain unaltered ; whence it follows that the mean 

 plane of revolution, which affbrds the maximum of pro- 

 jected areas, must pass through the centre of gravity of 

 the system, and remain always parallel to itself during the 

 motion of the system ; and that the sum of the areas, com- 

 puted for any plane perpendicular to this plane, must always 

 vanish. 



[Scholium. The whole of this elaborate demonstration 

 is rendered perfectly superfluous, if we exclude the distinc- 

 tion of absolute and relative motion from the definitions 

 relating to it (218, 224) : but it is satisfactory to find that 

 a complicated analysis is still true when applied to the test 

 of demonstrating by it a very simple proposition.] 



332. Theorem. The sum of the projec- 

 tions of the areas, described by the radii join- 

 ing each body of a system with each of the 



