506 XI. HYDRODYNAMICS. 



sphere, the - coordinate z l9 the velocity of efflux q at a point where 

 z = # 2 i g given "by 



43) a' 



or the velocity of efflux is equal to that acquired by a body falling 

 freely from a height equal to that of the free surface to the orifice. 

 This is Torricelli's theorem. 



190. Circulation. We define the circulation along any path 

 as the line integral of the resolved tangential velocity, 



B B 



44) <PAB = I q cos (q, els') ds = j (u dx -f v dy -f tv d#), 



A A 



corresponding to the circulation for displacement in 1 69. By Stokes's 

 theorem this is converted for a closed path into the surface integral 



45) 



()V 



over any surface bounded by the path. But this is by 26), 

 46) 2 / / {I cos (nx) + v\ cos (ny) -f J cos (ni)} dS 



= 2 I I G)cos(na))dS, 



that is, the circulation around any closed contour is equal to twice 

 the surface integral, over a cap bounded by the contour, of the resolved 

 normal vorticity. By the definition of ,??,, 26) 



identically, or the vorticity is a solenoidal vector. Accordingly by 

 applying the divergence theorem to a vortex -tube, or tube whose 

 generators are vortex -lines (lines whose tangents have the direction 

 of o), we find that the integral over any section of the vortex- tube is 



I 



& cos (no) dS 



L/ 



a constant for the tube. 



The fluid within any vortex -tube constitutes a vortex. If the 

 vortex is contained in a tube of infinitesimal cross -section S, the 



