346 Mr. G. H. Bryan on the Equations of 



and therefore by equating the two expressions for H we must 

 have 



£u + riv + &o + \p + iJLq + vr + 'F{K)=:-ttcpx- ■ • (22) 

 Since y m is the generalized velocity-component corre- 

 sponding to the momentum K m p, therefore 



^ = " x ; •'."'.' (23) 



Now H is a homogeneous quadratic function of the six 

 velocities (u . . . ,p . . .) and the momenta /cp ; therefore 



2H = 2m|5 + % K p ^5- = 2u|S -W. • (24) 

 Hence, from (21), 



2T=X^| FI +S^ % = ^P-^/)^-. . . (25) 



Qm ™ B^ ^^.Kp V y 



The portions of T and H which involve only the momenta Kp, 

 and are independent of the six velocities (u . . ., p . ..), must 

 arise from the terms 2k/?% in the above expressions (24) (25), 

 and must therefore be equal and of opposite sign in the ex- 

 pressions T and H respectively. Hence, since from (3) 



T=X / + X 1 + K, 



the portion of H which is independent of the six velocities 

 (u . . ., p . . .) must be — K ; so that 



H = X / + X 1 +(^-f-^y + ^ + X /? + ^ + vr)-K 



= T + (fu + ^ + ?i0 + \p+/^ + vr)-2K, . . (26) 



and therefore F(k P ) = —2K (27) 



The function F(/c/o) does not enter into the six equations of 

 motion of the solid, but its form requires to be determined if 

 we wish to reduce the equations of motion of the whole 

 system to the canonical or Hamiltonian form. 



The Generalized Velocities and Momenta. 

 10. Comparing (21) with (27), we see that 



2,/cpx = 2K — (f M + V v + tyo + \p + /J/ q + vr ), . (28) 

 Now equation (3) may be written in the form 



2 (Si + K) = -pjj^(«^ + . . . +p<f> p + . . . + 2*} K ) d B 



<,<f> u + . . . +p<f> p + ... + % K ^)d<r. (29) 



HJl^ 



