relative to Stationary Motions. 239 



Then T and U are the two quantities which Rankine has named 

 the actual and the potential energy, and whose constant sum is 

 the total energy, or briefly the energy of the system. If we de- 

 note the latter by E, the preceding equation reads : — 



T + U = E (3) 



If now, for the determination of the positions of the movable 

 points, instead of the rectangular coordinates any other variables 

 be introduced, which we will denote by q v q 2 , . . . q n , of course 

 the ergal U is to be regarded as a function of these variables. 

 As regards the other quantities occurring with the motion, and 

 the equations holding for it, the forms which they assume when 

 those general variables are employed are laid down by Lagrange 

 in his Mecanique Analytique. 



In order to ascertain what form the expression for the vis viva 

 takes, let us put for example, since the rectangular coordinates 

 of the points are to be regarded as functions of those general 

 variables, 



^/telJ &>>•••?«)• 





From this follows 





d^_4f_dq 1 ,4f_ d<h, 

 dt dq x dt dq% dt 



■ df dg n> 

 dq n dt' 



or, otherwise written, 







■ + dq n qn - 



■ (4) 

 In like manner can all the velocity-components of the movable 



points be expressed. As the differential coefficients —-, —-, . . .—— 



dq } dq Q dq n 



are functions of the n quantities q, the expressions of the velocity- 

 components contain the n quantities q and the n quantities q\ 

 and are, in relation to the latter, homogeneous of the first degree. 

 If we now imagine these expressions put in equation (2), we ob- 

 tain for the vis viva T an expression which also contains the 

 quantities q { , q 2 , . . . q n and q' v q' 2 , . . . q' n , and in relation to the 

 latter is homogeneous of the second degree. 



From the last-mentioned circumstance it follows, further, 

 that we can form the equation 



orn dH . dT . dT , 



or, using the sign of summation, 



