378 MATHEMATICAL APPENDICES 13* 



which express the fact that the collision makes no alteration in 

 either the total kinetic energy or in the motion of the centroid of 

 the two particles. 



For the solution we employ in this case too the method of the 

 calculus of variations, which may be here employed without 

 hesitation, as the components are variable magnitudes. But we 

 are also entitled to consider the components u, v, w as variable at 

 the same time that the components U, V, W are constant ; in this 

 case we limit our consideration to the cases of collision in which 

 from the original values u, v, w the same values U, V, W always 

 result. We thus obtain the equation 



= F(u. 2t v 2 , w 2 )SF(u lt v lt wj + F(u { , v lt io^F(u 2 , v 2 , w 2 ), 



and we have also the conditions 



= u^u } + v } cv l + w^Wi + u^ 

 =. ^u l + cu<2 



= %Vi + ^ 2 

 = ^W { + %W<2. 



By the former method of elimination we then at once obtain the 



known equations 



,, 1 dF(u,v,w) 



= - - - + 2km(v - ft) 



F(u, v, 10) dv 



= 



F(n, v, w) dw 

 in which both u lt v lt w l and w 2 , v 2 , w% are to be put for u, v, w. 



14*. Integration of the Equations 



The equations we have obtained hold good for all values which 

 we can assume for the components u l ,v l ,w l or u^v 2 ,w^ or in 

 general u nt v w w n , of the velocity of a molecule. We may then 

 take them as any variable magnitudes we like between the limits 



