234 SIR W. THOMSON ON VORTEX MOTION. 



But by (9) 



JJJx %dxdydz =JJJcxdxdydz -JJJx (| + ^dxdydz 



and by simple integrations, 



1 1 1 \ rT + -J-) dxdydz = f/x(v dxdz + w dx dy). 



Using these in (10), and altering the expression to a surface integral, as in 

 Thomson & Tait, App. A (a), we have 



fffu dx dy dz =ffx (u dydz + vdzdx + w dx dy) —fffcx dx dy dz 



= ffx1$d«-fffcxdxdydz (11), 



which clearly agrees with (7). 



When this mass is incompressible, we have c = o by the formula so ill named 

 the equation " of continuity" (Thomson & Tait, § 191), and we fall upon (8.) 



(lit 



The proper analytical interpretation of the differential coefficients -r , &c, 



and of the equation of continuity, when, as at the surfaces of separation of fluid 

 and solids, u, v, w are discontinuous functions, having abruptly varying values, 

 presents no difficulty. 



44. In the theory of the impulse applied to the collision (§ 29) of solids or 

 vortices moving through a liquid, the force-resultant of the impulse corresponds, 

 as we have seen, precisely to the resultant momentum of a solid in the ordinary 

 theory of impact. Some difficulty may be felt in understanding how the zero- 

 momentum (§ 4) of the whole mass is composed ; there being clearly positive 

 momentum of solids and fluids in the direction of the impulse in some localities 

 near the place of its application, and negative in others. [Consider, for example, 

 the simple case of a solid of revolution struck by a single impulse in the line of 

 its axis. The fluid moves in the direction of the impulse, before and behind the 

 body, but in the contrary direction in the space round its middle.] Three modes 

 of dividing the whole moving mass present themselves as illustrative of the dis- 

 tribution of momentum through it ; and the following propositions (§45) with 

 reference to them are readily proved (§§ 46, 47, 48). 



45. I. Imagine any cylinder of finite periphery, not necessarily circular, com- 

 pletely surrounding the vortices (or moving solids), and any other surrounding 

 none, and consider the infinitely long prisms of variously moving matter at any 

 instant surrounded by these two cylinders. The component momentum parallel 

 to the length of the first is equal to the component of the impulse parallel to the 

 same direction ; and that of the second is zero. 



II. Imagine any two finite spherical surfaces, one enclosing all the vortices 



