30 Prof. R. Clausius on the Theorem of the Mean Ergal, 



we thereby get, in addition to the foregoing, the following equa- 

 tions : — 



1 f/(U-T) . 1JE d\J 1 — r- lR \ 



v v 



I will here mention that in a recent memoir* I have developed 

 the equation 



2^ viv 2 <fy„ iv 2 ^ 

 Now, if the variables q l} q 2 , . . . q n are of such a nature that 

 they can be unambiguously determined from the position of the 



points, the mean value of the differential coefficient -~±jl±il in a 



dt 

 stationary motion is to be considered a vanishing quantity; 

 hence, if the mean values be taken, the equation will become 



1__ 1 rf(U-T) 



SM>2-25— *• ...... (?) 



The expression here standing on the right-hand side is the virial 

 referred to the variable q v . 



Now, in equations (6#), three other expressions occur which 



are also equal to ~P v qJ> an( ^ can therefore be regarded as expres- 

 sions of the virial. Besides it is to be remarked that these latter 

 are more convenient to use, inasmuch as they are products which 

 can be at once analyzed into two factors; while, the expression 

 in (7) being the mean value of the product of two variables, such 

 an analysis of it cannot be effected. 



If, in equation (5), we consider the terms which are affected 

 by the variatioas Sc ir $c, 2 , &c, understanding by jju one of the 

 indices 1, 2, &c, we can derive partial equations of the following 

 form :■ — 



K"K (8) 



Here the extremely slight difference, that the horizontal stroke 

 stands on the left side over the U only, and on the right side 

 over the entire differential coefficient, makes an important dif- 

 ference in the signification, and the meaning of the equation can 

 be expressed as follows : — When the mean ergal is considered as a 

 function of u lt xi 2 ., • .xi ni c 19 c 2 , &c, and these are differentiated 

 according to c^, we get the same quantity as when we differentiate 



* Pogg. Ann. Jubelband, p 411, equation (40); Phil. Mag. S. 4. vol. 

 xlviii. p. 1 (equation 40, p. 9). 



