4 Prof. A. Gray : Notes on Hydrodynamics. 



It will be noticed that this result is independent of the 

 position of the point P" with respect to which the angular 

 velocities of the particles have been taken. A small spherical 

 portion of the fluid with its centre within the infinitesimal 

 parallelogram, has a component of angular momentum about 

 the normal to the plane of dsds', of amount 



8 5 



Y* ?rpr cogs', 



where r is the radius of the radius and p the density of the 

 fluid. Thus we call co ss t the component of elemental angular 

 velocity of the fluid about the normal considered. It is an 

 affair of an element of the fluid mass, not of a particle. 



5. It is easy to extend the process by which (7) has been 

 established to show that the circulation round any closed 

 path drawn in the fluid is equal to twice the surface integral 

 of rotation of the fluid about the normals to the elements of 

 any surface of which the path is the bounding edge, and 

 indeed to prove Stokes's theorem connecting the line integral 

 of a directed quantity taken round the boundary of a surface, 

 with integral of the curl of the same quantity taken over 

 the surface. In what follows we shall assume the theorem 

 of circulation (not, however, of the constancy of circulation 

 for a closed path moving with the fluid). 



6. We may interpret the result stated in (5) and (6) 

 above in the following manner. First integrate along the 

 path of which PP' is an element from an initial point A to a 

 final point B. We obtain from (6) 



X A ~ —% 1 co gs q sin 6 ds — 1 



w 



X B -X A = -* \ co gs/ qsm6ds- \ ^ds'. . . (8) 



AB AB 



Now as q is the resultant velocity of the fluid at any point 

 P on the path AB, q' ( = qcos0) is the component at P 

 along the tangent drawn there to AB, while q sin 6 is the 

 velocity with which each particle P on the path AB is being 

 carried by the motion towards the right (see fig. 2) in the 

 plane of the diagram. The product q sin ds f is therefore 

 the rate at which an area of which ds' is an element of 

 boundary, and which is situated to the left of AB, is in- 

 creasing (or if the area is situated to the right of AB, is 

 diminishing) in consequence of the motion of ds' as a whole, at 

 right angles to itself, in the plane of the diagram. We see, 

 therefore, that the first term on the left is twice the rate at 

 which the surface integral of elemental rotation, taken over 



