2 ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



where H represents terms containing only the second differential 

 coefficients of A with respect to the coordinates, and the first differ- 

 ential coefficients of the coordinates with respect to the time. There- 

 fore, if we conceive of a variation affecting the accelerations of the 

 particles at the time considered, but not their positions or velocities, 



we have 



dA 



(3) 

 dB 



and, in like manner, 



etc. 



y 



Comparing these equations with (2), we see that when the accelera- 

 tions of the particles are regarded as subject to the variation denoted 

 by S, but not their positions or velocities, the possible values of Sx, Sy, 

 Sz are subject to precisely the same restrictions as the values of Sx, 

 Sy, Sz, when the positions of the particles are regarded as variable. 

 We may, therefore, write for the general equation of motion 



2[(X-m&)8x+(Y-my)8y+(Zmss)8z] = Q, (4) 



regarding the positions and velocities of the particles as unaffected by 

 the variation denoted by S, a condition which may be expressed by 

 the equations ^ = 0> 3 0> Sz = Q,) 



We have so far supposed that the conditions which restrict the 

 possible motions of the systems may be expressed by equations 

 between the coordinates alone or the coordinates and the time. To 

 extend the formula of motion to cases in which the conditions are 

 expressed by the characters = or ^, we may write 



I > [(X-mx)Sx+(Y-my)Sy+(Z-mz)Sz]^Q. (6) 



The conditions which determine the possible values of Sx, Sy, Sz 

 will not, in such cases, be entirely similar to those which determine 

 the possible values of Sx, Sy, Sz, when the coordinates are regarded as 

 variable. Nevertheless, the laws of motion are correctly expressed 

 by the formula (6), while the formula 



2[(X-mx)8x+(Y-my)8y+(Z-mz)8z]^Q, (7) 



does not, as naturally interpreted, give so complete and accurate an 

 expression of the laws of motion. 



This may be illustrated by a simple example. 



Let it be required to find the acceleration of a material point, 

 which, at a given instant, is moving with given velocity on the 



