Nature and their Mutual Dependence. 9 



particles, it also necessarily results from this that it is by no 

 means disagreeing with nature to apply the stated principle to 

 the propagation of heat in bodies, as, on the contrary, it leads to 

 truths proved by experience. 



Now I shall proceed to examine how the internal quantity of 

 energy contained in a fluid must vary when the pressure and 

 density of the fluid vary. 



Let dm be the element of the liquid mass m, whose particles, 

 according to the above, must be supposed to be in incessant in- 

 ternal vibration ; let the coordinates of the point of mass in 

 question, after the time t, be x, y, z, and let Xdm, Ydm, Zdm be 

 the moving forces on dm in the direction of the three rectangular 

 coordinate axes ; further, let the density at this instant for the 

 said point of mass in be p, and let p be the pressure on the unit 

 of surface ; if moreover the velocities of the element dm in the di- 

 rections of the three coordinate axes be denoted by 

 dx du dz • 



dt at at 



and if the increments of the velocities during the time dt are 

 equated to l( j dt} v r. dtj y dtj 



then, in conformity to the above, the increment of mechanical 

 energy which the element dm would have received during the 

 time dt if it had been perfectly free will be 



dm(Xdx + Ydy + Zdz). 



But, as the element dm is not perfectly free, in fact it only receives 

 an increment which may be represented by 



dm(u'dx -f v'dij + w'dz). 



During the 'time-element dt this element of mass consequently 

 loses some part of the mechanical energy which is really pro- 

 duced through the accelerating forces. If the energy which dm 

 loses during the time t is represented by q . dm, then the energy 

 lost in the time-element dt is equal to dq . dm, and thus we get 



dq . dm=[{X-u')dx+(Y-v')dy + (Z-w'yizJdm. . (11) 



But this internal energy dq . dm, which is imparted to the ma- 

 terial resistances by the element dm during the time dt, may be 

 put in a simpler form ; for, as is well known, we have 



dx dij dz .~=(X — u') dm, 

 dx dy dz . ~ = ( Y — v ! ) dm, 

 dx dy dz . ■£ .={Z — w')dm ; 



(12) 



