49] EFFECT OF ELIMINATION. 183 



' s = r 



2*'*' 



=i 

 on referring to tlie definition of the definitions of the 5 s 's, 138), 



159) S, = Q s , r 



Finally the terms from the lower right hand square, of m r rows 

 and columns gives us a quadratic function of the last m r velocities, 

 namely that part of 2T which originally depended on these velocities 

 and no others. This part we will call 2T a . We have therefore 



160) 2T = 2T tt + g.' (8. + e,\ 



5 = 1 



Now if we form the quadratic functions, with the coefficients R from 

 the determinant of equations 139), 



s = r t = r 



= -/, /, KitSt 



we may write equations 139) as 



lea) <?/=!-!! '(.- 1,2, ..,-), 



s s 



so that we may write 



s = r 



163) 2T = 2T a +2 (c, + S,) g - g) 



s = l s s 



But since (7, /S are homogeneous functions of c s , S s respectively, 



S=l * 8 = 1 



so that the above becomes, 



164) 2T=2T a + 2C-2S 

 But we also have 



so that the sums in 164) destroy each other, and there remains 

 165) T' = T a -S+C. 



