OF ENERGY IN A SYSTEM OF MATEBIAL POINTS. 



731 



of the first seven velocity-components are replaced by the differentials of the 

 seven constants. 



The functional determinant is found by differentiating the seven quantities 

 U n , V n , W n , F n , G n> H n and E with respect to the momenta m^, ra^, 

 .iv, ; m 2 u 2 , m^\, m a w a ; and m 3 u 3 . We thus obtain 



1, 0, 0, 0, z lt -y lt u, 



0, 1, 0, -z lt 0, x lt v, 



0, 0, 1, y -a;,, 0, w l 



1, 0, 0, 0, z,, -y,, , =A (63), 



0, 1, 0, -z,, 0, a:,, r , 



0, 0, 1, y,, x 3 , 0, w, 



1, 0, 0, 0, ,, -y s , -w 3 



which we may write A = a r 12 f 12 (64), 



where a = (?/, - y.) (z, - z 3 ) - (y, - y,) (z, - z 2 ) (65), 



or twice the projection on the plane of yz of the triangle whose vertices are 

 m lt m a , and m 3 , and 



or the rate of increase of the distance between m 1 and m 2 multiplied into that 

 distance. 



In a system composed of material particles, each component of momentum 

 is equal to the corresponding velocity-component multiplied into the mass of 

 the particle. We may therefore write ^ 1 = m 1 1 and so on, and since the masses 

 are invariable we may omit them from both members of equation (17), and write it 



dx{. . .dz n ' du^. . .dw n ' = dx^ ..dz n du. i ..,dw n (67). 



But d Ud Vd WdFdGdHdE = m* m? m 3 V M r' 12 du.'dv.'dw.'du.'dv^dw.'du,' 



dx 1 '.^dz n 'dv a '...dw n ' dx l ...dz n dv 3 ...dw n . 



Hence - , . y p- = m * m m ar r = ' ( 69 )' 



ll< \ I''-: //' .j U- ' U ' U //t-I //(.I //f-J Ur/ i" ' 1" 



and equation (29) becomes 



r ll )- 1 dv,...dw n (70). 



922 



