relative to Stationary Motions. 269 



fuller cxpres: 

 put 



•7T 

 the fuller expression ~ ,, and further, agreeably to equation (8), 



dt dq dq 

 Then follows 



w { ^ )=l S Siq ' +< S Sifj ~ d ^ s ' q - ■ • • {i3) 



An equation of this form holds for each of the n variables; and 

 if we imagine the sum formed from these n equations, we obtain 



Since the quantity T is a function of the 2n quantities q x , q. 2 , 

 ...q n and q' v q f 2 , . . . q' n , we can put 



dT dT 



an expression which contains the first two sums on the right- 

 hand side of our preceding equation. As regards the last sum 

 in that equation, if in U the quantities q } , q 2 , . . . q n were the 

 only variables, it could be replaced by 8 t \J. But as in U, by 

 hypothesis, other quantities c v c 2 , &c. occur, which, though in- 

 dependent of the time, yet may change their values on passing 

 from one motion to the other, therefore 



1 dq * dc 



By the employment of these two equations, (14) is transformed 

 into 



or, differently arranged, 



"dt"^ 1 ' 1 ' ~ dc 



-J>-£^8tf+s5?&. • • (16) 



This equation, multiplied by dt, then integrated from to /, 

 and finally divided by t, since h and k are the initial values of p 

 and q, takes the following form : — 



ij V-T)*= -^M=m + s i j'f &«. 



In the last term of the right-hand side, by making use of the 



