64.] GREEN S THEOREM. 59 



is - (0u + dx J dy dz. The sum of these is gj- dxdydz. 



Calculating in the same way the parts of the integral due to the 

 remaining pairs effaces, we get for the final result 



.&amp;lt;bii d . &amp;lt;bv d.&amp;lt; 



Hence {17) simply expresses the fact that the surface-integral (18), 

 taken over the boundary of S, is equal to the sum of the similar 

 integrals taken over the boundaries of the elementary spaces of 

 which we have supposed S built up. 



We may interpret (17) by regarding u, v, w as the component 

 velocities of a continuous system of points filling the region S, and 

 supposing (/&amp;gt; to represent some property (estimated per unit 

 volume) which they carry with them in their motion. The surface- 

 integral on the left-hand side of (17) expresses then the amount 

 of (/&amp;gt; which enters S in unit time across its boundary ; whilst the 

 above investigation shews that (19) expresses the rate at which 

 the property &amp;lt; is being accumulated in the elementary space 

 dxdydz. The theorem then asserts that the total increase of &amp;lt; 

 within the region is equal to the influx across the boundary. A 

 particular case is where u, v, w are the component velocities of a 

 fluid filling the region, and &amp;lt; is put = p, the density. See Art. 12. 



Corollary 1. Let = 1; the theorem becomes 



dv dw\ 7 , , 

 + Ty + IT, ) dxd ^ 



If u } v, lu be the component velocities of a liquid filling the region 

 S; the right-hand side of this equation vanishes, by the equation 

 of continuity; so that 



ff 

 JJ 



which expresses that as much fluid leaves the region as enters it. 



Corollary 2. Let u, v, w=-, -7, - , respectively, where 



^r is a function which, with its first differential coefficients, is 

 finite and continuous throughout S. Then 



