and Us Application to the Molecular Motions of Gases. 27 



system can be represented by a function of these variables and 

 their differential coefficients, taken according to the time, which 

 we will designate by q' v q'%, . . . q' n , which function is homoge- 

 neous of the second degree, and therefore satisfies the equation 



2T- dT a' + ' dT a> 4- + dT a< 



If, for shortness, we put 



?>=!?; ' W 



v denoting any one of the indices from 1 to n } the equation 

 becomes 



or, employing the sign of summation, 



2T=Zpq' . (2) 



We may now suppose that another, infinitesimally different 

 stationary motion takes the place of the one previously considered. 

 The difference may be occasioned either by the initial positions 

 and velocities of the points not being precisely identical with 

 those in the original motion, or by the function U, representing 

 the ergal, having a slightly different form. In order to express 

 the latter more simply, we will assume that the function U con- 

 tains, besides the variables q v y 2 , . . . q n , also one or more quan- 

 tities which are constant during each of the two motions, but 

 have not quite the same values in the one motion as in the other. 

 The primitive values may be designated by c v c 2) &c, and the 

 changed values by c L +Sc v c 2 + Sc 2> &c. 



If we now suppose the mean value of the ergal to be formed 

 during each of the two motions, one of these two mean values is 

 somewhat different from the other ; and we call their difference 

 the variation of the mean ergal. It is this variation which is to 

 be determined by my equation, as it is brought into relation 

 with other variations. 



To make the equation as readily intelligible as possible, it shall 

 be first presupposed that both in the original and in the altered 

 motion the variables q v q 2 , . . . q n accomplish their variations 

 periodically. We will write i X) z 2 , . . . i n for the time-intervals 

 which serve as the periods of the individual variables in the ori- 

 ginal motion, and i i -{-Bi 1 , i 2 + 8i 2 , . . . i n + $i n for the same in the 

 altered motion. Further, of every variable quantity in the 

 course of the motion, we will distinguish the mean value from 

 the variable values by putting a horizontal line over the symbol. 



