456 Dr. L. Natanson on the Kinetic Interpretation 



the components of the mean (or " molar ") velocity within an 

 element dxdydz of volume; ndxdydz may represent the 

 number of molecules within that element. The mass of a 

 molecule being M, p = M?i being the density of the medium, 

 the kinetic energy of a molecule is 



|M{(u+?) 2 +(,+7 ? ) 2 + («6-+r) 2 }; • • • (i) 



and the total kinetic energy of a portion of the medium con- 

 sists of the two following parts : — (1) the kinetic energy of 

 the visible motion, 



K = ±$ p (u 2 + u 2 + iv*)d,vdydzi ... (2) 



it is this energy that, in Hydrodynamics, is taken into account ; 

 (2) the molecular or heat-energy, 



& = ^§p(J>W+?) dxdydz. ... (3) 



In these expressions the integrations are supposed to be per- 

 formed throughout the volume occupied by the medium ; 

 and f 2 and similar symbols represent the mean values of % l 

 &c. for the molecules within an element. We have 



|=0; ,7 = 0; ^ = (4) 



At the point ^c, y, z) we have the normal component 

 pressures, 



Pxx-pi 2 ; Pyy=pri 1 ; pzz-pK 2 ) • • • ( 5 ) 



and the tangential, 



Pyz—Pzy=pvKl Pzx=Pxz=p$i ', Pxy=Pyz = P%V- • (6) 



Let Q be any property of the molecule which can be 

 expressed as a function of (w + f), {v + ri), and (w + ?). Then 

 writing d/dt for the actual or total variation of Q, and S/St 

 for any alteration of Q that can be due to the mutual inter- 

 ference of molecules, 



=„f +P xf +P Yf +P Zp; . (7) 



X, Y, Z denote here the components, at (x, y, z) , of accelera- 

 tion due to external forces. From this fundamental equation 

 (7) the equation of continuity, as well as the equations of 

 motion, i -^ __ -n -s 



immediately follow. Let us now put 



Q=( w + £) 2 +( y + 7;) 2 -f(™ + t) 2 ; ....<<)) 



