SIR WILLIAM THOMSON ON VORTEX MOTION. 247 



P : , respectively; and 2* denoting integration along the arc from P 1 to P 2 . Let 

 now P 2 be moved forward, or P x backward, till these points coincide, and the 

 arc P X P 2 becomes the complete circuit; and let 2 denote integration round the 

 whole closed circuit. (6) becomes 



dt 



= (7); 



and we conclude that 2£Ss remains constant, however the tube be varied. This 

 is the proposition to be proved, as the "average velocity" referred to is found 

 by dividing 2(£Ss) by the length of the tube. 



59. (d). The tube, imagined in the preceding, has had no other effect than exert- 

 ing, by its inner surface, normal pressure on the contained ring of fluid. Hence 

 the proposition* at the beginning of § 59 is applicable to any closed ring of fluid 

 forming part of an incompressible fluid mass extending in all directions through 

 any finite or infinite space, and moving in any possible way; and the formula? (5) 

 and (6) are applicable to any infinitesimal or infinite arc of it with two ends not 

 met. Thus in words — 



Prop. (1.) The line-integral of the tangential component velocity round any 

 closed curve of a moving fluid remains constant through all time. 



And, Prop. (2), The rate of augmentation, per unit of time, of the space 

 integral of the velocity along any terminated arc of the fluid is equal to the 



* Equation (6), from 'which, as we have seen, that proposition follows immediately, may he 

 proved with greater ease, and not merely for an incompressihle fluid, hut for any fluid in which the 

 density is a function of the pressure, by the method of rectilineal i-ectangular co-ordinates from the 

 ordinary hydrokinetic equations. These equations are— 



Dm dzr 



Did dsr 



Dzo d-sr 



Dt ' dx ' 



Dt ~~ dy' 



Dt ~ dz 



if — denote rate of variation per unit of time, of any function depending on a point or points moving 



_L/fc 



with the fluid; and -zr = / -j § denoting density. In terms of rectangular rectilineal co-ordinates 



we have 



$s = uSx -f- vhy + wBz . 



Hence 



D(§&) Du D&* „ 



— ^rjbx + U -jrr- + &C. 



D* _ Dt "* T " D* 



Now 



DBx DSy DSz . 



~Df= Bu ' -DT = Bv > and^-^Sw;. 



These and the kinetic equations reduce the preceding to 



^^ ) = uBu + v Bv + wBw-^-^8y-^Bz = S[U^ + ^ + ^)-^\ • ( 8 ) i 

 whence, by 2 integration, equation (6) generalised to apply to compressible fluids. 



