applicable to Heat. 127 



duration of a period may be taken as the time t ; but for irregular 

 motions (and, if we please, also for periodic ones) we have only 

 to consider that the time t, in proportion to the times during 

 which the point moves in the same direction in respect of any 

 one of the directions of coordinates is very great, so that in the 

 course of the time t many changes of motion have taken place, 

 and the above expressions of the mean values have become suffi- 

 ciently constant. 



The last term of the equation, which has its factor included 

 in the square brackets, becomes, when the motion is periodic, 



d(a?) 

 = at the end of each period, as at the end of the period , ■ 



fdix*) \ . 



resumes the initial value ( ,, J . When the motion is not 



\ at J 



periodic, but irregularly varying, the factor in brackets does not 

 so regularly become =0; yet its value cannot continually in- 

 crease with the time, but can only fluctuate within certain limits ; 

 and the divisor /, by which the term is affected, must accord- 

 ingly cause the term to become vanishingly small with very 

 great values of t. Hence, omitting it, we may write 



2\dt) ~~ 



-}X*. 



As the same equation is valid also for the remaining coordi- 

 nates, we have 



![©HI) + (S)>-^ s ^w, 



or, more briefly, 



m 



and for a system of any number of points we have the perfectly 

 corresponding one 



fn-s 



S^e-iSfXa+Yy+Z*). 



Hence our theorem is demonstrated ; and at the same time it is 

 evident that it is not merely valid for the whole system of mate- 

 rial points, and for the three directions of coordinates together, 

 but also for each material point and for each direction separately. 



