8 PROFESSOR ANDREW GRAY, 



If, according to the usual notation, we put 



these equations become 



dw dv _du dw os-_3'"' du 



dp dz' dz dx dx dy 



°£ + 2w v -2vZ = 0, ^ + 2^-2^ = 



• (18), 



which may be regarded as the Cartesian equations of motion of the fluid. These 

 equations are given in Lamb's Hydrodynamics, 3rd edition, § 146, but are derived 

 directly from the ordinary Eulerian equations of motion. 



14. The proof of equation (15) deserves a little attention. We can show that the 

 equation holds, and also make it clear that w ss , is an angular velocity of rotation — 

 a point that is not always brought out in the kinematics of vortex-motion — by pro- 

 ceeding as follows. The average velocities along the four sides of the small parallelo- 

 gram of the figure may be taken to be the velocities of the fluid in these directions at 

 the middle points of the sides. These are, for the sides PQ, QR, RQ', Q'P, 



1 ^1 7 I dl l 7 M T 



9 + hfods, 2+ & *> + *&*»> 



/ ,d</ 3<7 A f . , W , A 



Multiplying the first and third of these by ds, and the second and fourth by ds', and 

 adding, we obtain 



3s 3s'/ 



Thus if q c be the component of velocity along the boundary of the parallelogram 

 at any element dc, we have for the parallelogram 



I qjlc = (JL — -2, )dsds. 

 J ic \ds dsj 



But the angular velocity of a fluid particle at dc about a point, P" say, in the 

 plane of ds and ds' and within the parallelogram, is qjp, where p is the length of 

 the perpendicular from P" on the side upon which dc is situated. Thus 



{'h p dc=( d l-^\d,ls. 

 J j? \3s ds'J 



If co„, be the mean angular velocity for particles at different points of the periphery 

 of the parallelogram, the integral on the left is evidently 2w sg -dsds' sin 6. Thus 

 we obtain, as stated in (15), 



