502 Dr. L. Natanson on the Kinetic Energy 



an element, and X, Y, Z the components of acceleration due 

 to external forces at the point (#,?/, z). We will write dQ/dt 

 the total or actual variation of Q ; and by BQ/Bt we will 

 represent such variation of Q as can be due to the mutual 

 interference between molecules. Then (Maxwell, " On the 

 Dynamical Theory of Gases/' Scientific Papers, ii. p. 26), 



If we give to Q in this equation consecutively such signifi- 

 cations as are consistent with definition and then eliminate 

 terms including X, Y, Z and S/8t, a set of propositions will 

 be obtained, constituting what may be called a Kinematical 

 Theory of Fluids, a theory of rather high degree of gene- 

 rality which must not be confounded with special molecular 

 theories of usually very hypothetical character. It is with 

 Hydrodynamics that the kinematical theory seems to be most 

 intimately connected, the fundamental hydrodynamical equa- 

 tions (or possibly some generalizations thereof) being simple 

 deductions from the equations of that theory. 



2. In equation (1) put 



Q=(u + f){(w + f)H(»+^) 2 +> + £l s }; . ■ (2) 



and let us write for brevity, 



iW+¥W)=r x (3) 



Neglecting small terms we obtain 



dt 



O , To 



+ ^- {p? 2 (3^ 2 + v 2 + u?) + p?(£ 2 + 1 + ? 2 ) } 

 ox 



+ A (2uvpv*) + |; (2mvpf) - 2pu |? + /> ^f 



+ P X(3u 2 + v 2 + v? + 3p + ^+P) + ZpYuv + 2pZuw. (4) 



We shall simplify this equation, being satisfied with a first 

 approximation . Put in (1) Q = (u + J) 2 ; put, again, Q = u + f , 

 multiply by 2w, and to a first approximation it follows, 



*f +<*£-** ■•■•<*> 



