158 VORTEX MOTION. [CHAP. VI. 



Now -x , the rate at which dx is increasing in consequence of 



the motion of the fluid, is evidently equal to the difference of 

 the velocities parallel to x at its two ends, i.e. to du; and the 



value of ~ is given in Art. 6. Hence, and by similar con- 



ot 



siderations, we find 



;r- (udx + vdy + wdz) - d V + udu + vdv + wdw. 

 Integrating along the line, from A to B, we get 



or, the rate at which the flow from A to B is increasing is equal to 

 the excess of the value which J (f V \ has at B over that 

 which it has at A. 



This theorem, which is due to Thomson, comprehends the 

 whole of the dynamics of a perfect fluid in the general case, as 

 equation (3) of Art. 25 does for the particular case of irrotational 

 motion. For instance, equations (26) of Chapter i. may be 

 derived from it by taking as the line AB the infinitely short 

 line whose projections were originally da, d~b, dc, and equating 

 separately to zero the coefficients of these infinitesimals. 



The expression within brackets on the right-hand side of (24) 

 is a single- valued function of x, y, z. It follows that if the in 

 tegration on the left-hand side be taken round a closed curve, 

 (so that B coincides with A,} we have 



dt 



l(udx + vdy + wdz) = 0, 



or, the circulation in any circuit moving with the fluid does not 

 alter with the time. See Art. 59. 



Applying this theorem to a circuit embracing a vortex-tube we 

 ;find that the strength of any vortex is constant. 



Also, remembering the formula given in Art. 39 for the cir 

 culation in an infinitesimal circuit, we see that if throughout any 



