of the Mechanical Theory of Heat. 167 



ordinate are valid also for the y and z coordinates, viz. : — 





-3^=KltHDW> . . . (in 

 -£«~*<$)'+(g)V • • ^ 



Adding these three equations, and at the same time taking 

 into consideration that 



(dx\* (dy\* , /<fc\ 2 a 



U) HI) + W=^ ■■ • •• ( J5 ) 



in which v signifies the velocity of the point, the result is : — 



" (Sfto+g^+g&J^i^+Wlog,-. (16) 



If we multiply this equation by the mass m of the material 



a"*x d*y 

 point, we can introduce, instead of the products m>jj%i m ~nr> 



d 2 z 

 and m ~j, the three components (taken in the directions of the 



coordinates) of the force operating on the point, which may be 

 denoted by X, Y, and Z, thus : — 



- (X8x + YSy + Z&?) = tt Sv 2 + mv*S log i. . ( 1 7) 



In reference to the force operating on the point, we have pre- 

 supposed that its three components may be represented by the 

 partial differential coefficients, taken negatively, of a function of 

 the coordinates of the point. If, for the original motion, we de- 

 note this function (which we call the ergal of the point) by U, 

 we can give to the preceding equation the following form : — 



^$ x + d V B+ dJl Sz= ™ Bv 2 + mvmi m m (18) 

 dx ay * dz 2 v ' 



or, more briefly, 



8U=^&? + wn?31ogf (19) 



5. In this equation we must first consider the expression SV. 



In every case in which, with the altered motion, the ergal is 

 still represented by the same function U as with the original 

 the quantity SU (the alteration of the mean value of the ergal) 

 expresses the work done in the transition from the one stationary 

 motion to the other. If, then, as we have done above in the 



