234 NOTES. 



kinematical ; thus we cannot assert that the model will, with the same 

 internal forces, work similarly to the original. 



Let u, v, w be the component velocities of the fluid, as above 

 denned, at the point (x, y, ), u + a, v + (3, w + y those of a particular 

 molecule of mass m in the neighbourhood of that point. Let the symbol 

 2 denote a summation extending through unit volume ; more precisely, 

 let 2 denote the result of a summation through unit volume on the 

 supposition that the properties of the medium are uniform throughout 

 this volume, and the same as at (x, i/, z). We have, then, in virtue of 

 the definitions above given 



*n = p (1), 



Sma^O, 2m{3=0, 2 t my=0 (2). 



If we consider a small closed surface moving with the fluid, the 



.9 d d d d 



symbol^-, ^ T t + u ~T +V cT +w rf~&amp;gt; applied to any function F, ex 

 presses the rate of variation of F considered as a property of the 

 included space. On account of the continual exchange of molecules 

 between this space and the surrounding region, this is not necessarily 

 the same thing as the rate of variation of F considered as a property of 

 the matter which happens to occupy this space at the instant in ques 

 tion. Hence we cannot, as in Art. 6, accept 



du 

 mass x 



ct 



as a complete expression for the rate of increase of the ^-momentum of a 

 moving element. A correction is rendered necessary by the passage to 

 and fro of molecules carrying their momentum with them across the 

 walls of the element. To form the equations of motion it is better, 

 then, to have recourse to the second method, explained in Art. 12, viz. 

 to fix our attention on a particular region of space, and study the 

 changes produced in its properties as well by the flow of matter across 

 its boundaries as by the action of external forces. 



Let us take then a rectangular element of space dxdydz, having its 

 centre at (x, y, z). Let Q denote any property of a molecule which it 

 can carry with it in its motion*. The total rate of increase of the 

 amount of Q in the above space is expressed by 



.3Qdxdydz ,.: (3). 



* The following investigation is substantially that given by Maxwell, /. c. 

 Art. 12. 



