10 Prof. R. Clausius on different Forms of the Virial. 



angles made by such right lines; or it may be ambiguous — 

 which is the case with the angles themselves, since to one direc- 

 tion an infinite number of angles belong, which differ from one 

 another by 27r. In the former case ^pq is a quantity the value 

 of which, with a stationary motion, varies only within certain 

 limits , and accordingly the mean value of the differential coeffi- 

 cient, taken according to time, of this quantity may at once be 

 regarded as vanishing and be omitted from the above expres- 

 sion. In the latter case, on the contrary, the mean value of that 

 differential coefficient does not necessarily vanish, and hence it 

 must remain in the expression for further consideration. 



Should the variables q l3 q q , q s , &c. not be all independent, 

 but connected with one another by certain condition-equations, 

 then we can, notwithstanding, obtain equations similar in form 

 to (40) by employing Lagrange's indeterminate coefficients. 

 Let, namely, 



&c. 



be the given condition-equations, we form instead of (38) the 

 following equation, 



where p, a, &c. are indeterminate coefficients ; and this equation 

 is to be resolved, in the usual way, into as many partial equa- 

 tions as there are variations. The partial equation correspond- 

 ing to the variable q v is then 



dU dT , dd> dty o 



whence results 



_ tf(T-U) , J<j> Jf 



y " = -^r +f> ^ +ff ^ +&c - ■ • (42) 



By the insertion of this value of p' v equation (37) is changed 

 into 



FvHv L dq v H dq v dq v l Hv dt K ' 



As many equations of this form are obtained as the given vari- 

 ables q v q q , q 3 , &c. ; and the work can be supplemented by eli- 

 minating from them the indeterminate coefficients. 



It is thus shown in a general way how the equations which 



