232.] GENERAL PRINCIPLES. 



similarly by dS y , dS z the corresponding sums of the projections 

 on the other co-ordinate planes. Then the equations (10) can 

 be written in the form 



2S X =C V 2S y =C 2 , 2S Z =C 3 . (I I) 



Hence the proposition of Art. 230 might be called the principle 

 of the conservation of sectorial velocities ; it is more commonly 

 called the principle of the conservation of areas. 



The equations (11) can be integrated again and give, if the 

 sectors be measured from the positions of the radii vectores at 

 the time t=o, 



232. If the radii vectores be projected on any plane through 

 the origin whose normal has the direction cosines a, fi, y, the 

 sum of the elementary sectors described in this plane, each 

 multiplied by the mass, will be 



hence S 



On the other hand, by (10), the angular momentum of the 

 body about the normal of this plane has the expression 



a + <$ + C s y, as it must be equal to the sum of the pro- 

 jections on this normal of the angular momenta about the 

 axes of co-ordinates, which can be regarded as vectors laid 

 off on these axes. 



Now it is easy to see that this angular momentum C^+CJS 

 -\-C 3 y, and hence the quantity 5 at a given time /, is greatest 

 for the diagonal of the parallelepiped, whose edges are equal 

 to C lt C z , C 3 along the axes, i.e. for the normal to the plane 



Cjx+C^y+C^ = o. (12) 



For, the direction cosines of this normal are 



', where D= V\ 2 4- C+C ; and the quan- 

 tity C^a. 4- C^ft 4- QY can be put into the form 



