OF THE MOTION OF A SYSTEM. 219 



For if the area be resolved into elementary rectangles, 

 the breadth of each, parallel to the common section of the 

 planes, being the same in the projection as in the original, 

 the length of the projection will be to that of the original 

 as the cosine of the inclination to the radius; and the 

 whole areas will be in the same constant ratio as their 

 elements. 



Corollary. Hence the sum of the squares of the 

 sines of the angles, formed by the three faces of the 

 parallelepiped with the section, is equal to the square of 

 the radius, or unity.] 



330. Theorem. For every independent 

 system of bodies, a fixed plane may be deter- 

 mined, with respect to which the sum of the 

 projections of all the areas, described by the 

 revolving radii, multiplied by the masses of 

 the respective bodies, is the greatest possible ; 

 and for every plane perpendicular to which, 

 the sum of the projections vanishes. 



" By taking the fluxions of the equations for the values 

 of a:,,,, y^^^y and z,,/' [, the angles remaining constant], " and 

 substituting c, q', and c'\ for 



2m — ^-TT-—y Im -jz , and Xm ^ — tt-~> we 



d^ d^ at 



obtain 



X dz ~~~jz djT 

 Dm '" "' '" ' — ^=:c sin decs ?> 4- c' (sin >^ sin ^ + cos d 



cos \^ cos (p) -{- c"(cos 4^ sin (p — cos d 

 sin -^ cos ^) ; 



