119-120] IMPULSE IN TERMS OF KINETIC ENERGY. 



175 



Since the variations Aw, Aw, Aw, Ap, A^, Ar are all independent, 

 this gives the required formulae 



dT dT fc dT\ 



dw' 



(3). 



dT 



V = 



dT 

 dr 



It may be noted that since f , t], f, . . . are linear functions of 

 u, v, w,..., the latter quantities may also be expressed as linear 

 functions of the former, and thence T may be regarded as a homo- 

 geneous quadratic function of f, rj, \, //,, v. When expressed in 

 this manner we may denote it by T'. The equation (1) then 

 gives at once 



whence 



drj 



_dT 

 = df 



dT' 



^ = dX 



" 



dT' 



dT 7 



(4), 



formulaB which are in a sense reciprocal to (3). 



We can utilize this last result to obtain another integral of the 

 equations of motion, in the case where no extraneous forces act, in 

 addition to those obtained in Art. 117. Thus 



___ 

 dt ~ df 



-.*+... + .. 



dX 

 dX 



... + ..., 



which vanishes identically, by Art. 117 (1). Hence we have the 

 equation of energy 



T'=const (5). 



120. If in the formula (3) we put, in the notation of Art. 118, 



rp rri rn 



it is known from the dynamics of rigid bodies that the terms in T! 

 represent the linear and angular momentum of the solid by itself. 



