48] ELIMINATION OF VELOCITIES. 177 



whose variation appears in Hamilton's Principle, the Hamiltonian 

 function H=T J r W. The transformation of Routh and Helmholtz, 

 instead of eliminating all the m velocities q', eliminates a certain 

 number, which we will choose so as to replace those having the 

 suffixes 1, 2, . . . r, by the corresponding momenta, but to retain the 

 velocities with suffixes r + 1, . . . w, in the equations. This trans- 

 formation, while it may be made in the general case, is of particular 

 advantage where the eliminated velocities are cyclic and the corre- 

 sponding momenta constant, as in the case just described. 



The equations 53) 39 for the elimination become by trans- 

 position 



138) 



Qriqi + GrSffi + '+ Qrrq'r = Pr ~ (fc.r + ltfr'+l +' ' ' + Crmffi). 



It will be convenient to write the right hand members above, 



Pi-Si, ...pr-Sr. 

 Let the solutions of equations 138) be 



&' = -Rll (Pi - $) + R** (P-2 ~ SJ + + Rlr (Pr ~ &), 



139) 



qr' = Rrl (Pi ~ S t ) + Bri (pi - S 2 ) + + Err (p r ~ &), 



where the J^'s are the quotients of the corresponding minors of the 

 determinant 



Qn> 612? Q 



Qrl) Qr2, - Qrr 



by the determinant itself, and, like the 's, are functions of the 

 coordinates only. 



Introducing the values 139) for the q"s into the kinetic energy, 

 the latter becomes a function of the velocities q r +i, - - q and of 

 the momenta p l9 .. .p r . It is a homogeneous quadratic function of 

 all these variables, but not of the p's or g"s considered separately 

 on account of product terms, such as p s q t ' which are linear in terms 

 of either the p j s or q"s. The function I thus transformed has lost 

 its utility for Lagrange's equations, but may be replaced by a new 

 function, as follows. 



Let us call the function T expressed in terms of the new 

 variables T'. We have thus identically 



140) T(q l9 q 2 >... q m , &', &', ...$!). 



WEBSTER, Dynamics. 12 



