10 ON THE FUNDAMENTAL FORMULA OF DYNAMICS. 



increments p\ x', ij f . z' t and subtracting the original value of the 

 expression from the value thus modified. Now, 



But since p, x' y y' t z' are proportional to and of the same sign with 

 possible values of Sp, 8x, Sij, 8z, we have, by the general formula of 

 motion, ^ (p-^ _ 2 [ m (&g' + yf + %#)] < Q. 



The second member of the preceding equation is therefore negative. 

 The first member is therefore negative, which proves the proposition 

 with respect to (15). The demonstration is precisely the same with 

 respect to (13) and (14), which may be regarded as particular 

 cases of (15). 



To show the same with regard to (16) and (17), we have only to 

 observe that the quantities affected by 3 in these formulae differ from 

 those affected by the same symbol in (14) and (15) only by the terms 



^(Xx+Yy+Zz) and (Pp), ? 



which will not be affected by any change in the accelerations of 

 the system. 



When the forces are determined by the configuration (with or 

 without the time), the principle may be enunciated as follows: The 

 accelerations in the system are always such that the acceleration of 

 the rate of work done by the forces diminished by one-half the sum 

 of the products of the masses of the particles by the squares of their 

 accelerations has the greatest possible value. 



The formula (17), although in appearance less simple than (15), not 

 only is more easily enunciated in words, but has the advantage that 



the quantity -ji(Pp) is entirely determined by the system with its 

 (Jut 



forces and motions, which is not the case with $(Pp). The value of 

 the latter expression depends upon the manner in which we choose to 

 represent the forces. For example, if a material point is revolving in 

 a circle under the influence of a central force, we may write either 

 Xx+Yy+Zz or Rr for P'p t R and r denoting respectively the force 

 and radius vector. Now Xx+Yy+Zz is manifestly unequal to Rr. 



But Xx+Yy + Zz is equal to Rr, and -^-(Xx+Yy + Zz) is equal to 



j CLv 



B<*> 



It may not be without interest to see what shape our general 

 formulae will take in one of the most important cases of forces 

 dependent upon the velocities. If a body which can be treated as a 

 point is moving in a medium which presents a resistance expressed by 



