SIR WILLIAM THOMSON ON VORTEX MOTION. 259 



2 denoting summation for the several barriers if there are more than one. 

 According to the general hydrokinetic theorem for irrotational motion [§ 59 (6) 

 compare with § 31 (5)], with <p expressed in terms of the co-ordinates of a point 

 moving with the fluid, we have 



,=-g+w m- 



Now, let us suppose the pressure to be impulsive, so that there is infinitely little 

 change of shape either of the bounding surface or of the barriers during the time fdt. 



This will also imply that -^ is infinitely great in comparison with \<f ; so that 



*=-£ <» 



And according to the notation of § 57 we have 



N = np (9). 



Also k is constant over each barrier surface. 

 Hence (6) becomes 



^\=Jdt^JT^ri 9 da^tkJfii 9 d^ . . . (10). 



64. (b). The initiating motion of the bounding surface and the pressures on the 

 barriers may be varied quite arbitrarily from the beginning to the end of the 

 impulse ; so that the history within that- period of the acquisition of the pre- 

 scribed final velocity may be altogether different, and not even simultaneous, in 

 different parts of the bounding surface. Thus k x and k 2 may be quite different 

 functions of t ; provided only fk x dt and fk 2 dt have the prescribed values, which 

 we shall denote by K and fc 2 respectively. 



(a). But, for one example, we may suppose <p to have at each instant of fdt 

 everywhere one and the same proportion of its final value ; so that if the latter 

 denoted by <I>, and if we put 



-§ = ™ (11). 



m is independent of co-ordinates of position, but may of course be any arbitrary 

 function of the time. Hence, observing that 



fd 



11 ill = « 



as the final value of m is 1, (10) becomes 



W = kUf&bQda + %kff*®d?~] .... (12). 

 (d). The second member of this equation doubled agrees with the two equal 



