118 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 



with respect to the corresponding velocity. The equation 38) now 

 says that the kinetic energy is one -half the geometric product of the 

 velocity and momentum systems. Thus we have perfect analogy with 

 the last two equations of 27. 



We shall hereafter denote the momentum belonging to q s by p s 

 and effecting the differentiation of 35) we have 



O Q 



or every generalized momentum -component is a linear function of 

 the velocities, the coefficients being the inertia -coefficients Q rs . 



Let us now find the component of the effective forces according 

 to q s , the effective forces being defined by the system of products, 

 for each particle, of mass by acceleration, 



dx[ dyl del 



We have 



to transform which we make use not only of 51), but of a relation 

 obtained as follows. Differentiating 50) by q s) 



h 



Using these results in 54), we obtain for the right-hand member, 



- E ; ) - *%] - iA k |* 



and with similar results for y and 8j summing for all the particles, 

 we have for the component of the effective forces of the system, 





d 



dt\dq' 



Putting the effective force equal to the applied force we have 

 Lagrange's equation 47) by direct transformation. The equation of 

 d'Alembert's principle thus becomes in generalized coordinates 



56) 



s = l 



