126 THEORY OF NEWTONIAN FORCES. [PT I. CH. III. 



Suppose that instead of reciprocating with regard to all the 

 velocities q as in Hamilton's transformation, we do so with regard 

 to only a number r of them which we will choose so that they shall 

 have the indices from 1 to r, while the q"s with indices r + 1, ... n, 

 remain in the reciprocal function, and with all the coordinates q 

 play the part of the variables z in 63. Then calling the negative 

 of the reciprocal function 



(1) T=T-i.q s 'p s , 



1 



we have 



dT dT 



= , for s=l, 2, ... n, 



( 2 ) fy* 9f 



dT dT 



w=*Z'** 8ssr + 1 ""*' 



and 



dT , dT f -to 



*-&' -?.=^.fo"=i.2,...r. 



Replacing T in Lagrange's equations by T, we obtain 



d_(W\ ST_ dW 

 '~- "* 



so that we may use for the suffixes corresponding to the un- 

 eliminated velocities Lagrange's equations, using the function, 



4> = T- W 



instead of the Lagrangian function 



L = T-W, 



and obtaining 



For the suffixes corresponding to the eliminated velocities we must 

 use the Hamiltonian form of the equations 



(6) - 



dt s 



dq 8 d(T-W) f 

 --ff= v a -, for s=l, 2 ... r. 

 dt dp s 



If r = 7i, T becomes T, and we have the complete Hamiltonian 

 form, 64 (4). 



