260 SIR WILLIAM THOMSON ON VORTEX MOTION. 



second members of (7) § 57 with <p and <p each made equal to 0. And the first 

 member of that equation becomes twice the kinetic energy of the whole motion. 

 Hence, when ?>' = ?>, and V 2 <p = 0, (7) of § 57 expresses the equation of energy 

 for the impulsive generation, of the fluid motion corresponding to velocity potential 

 <p, by pressures varying throughout according to the same function of the time ; 

 the first member being twice the kinetic energy of the motion generated, and the 

 second twice the work done in the process. 



64. (e). As another example, let us suppose the initiating pressures to be so 

 applied as first to generate a motion corresponding to velocity potential <p, and 

 after that to change the velocity potential from <p to <p + <p, denoting by <p and <p 

 any two functions, such that <p + <?' = <!>', and each fulfilling Laplace's equation: 

 and let the augmentation from zero to p, and again from <p to <p + <t> be uniform 

 through the whole fluid. The work done in the first process, found as 

 above (12), 



hlff^da + tKfPKpch-] .... (13), 



if k v k 2 , &c, denote the cyclic constants relative to <p, as ft x , fc 2 , &c, relatively to 

 <I>, and the additional work done in the second process, similarly found, is 



(/). Now, as we have seen (§ 63) that the actual fluid motion depends at 

 each instant wholly on the normal velocity at each point of the bounding surface 

 and the values of the cyclic constants, it follows that the work done in generating 

 it ought to be independent of the order and law, of the acquisition of velocity 

 at the bounding surface, and of the attainment of the values of the several cyclic 

 constants. Hence, the the sum of (13) and (14) ought to be equal to (12). But 

 if, for O in (12) we substitute <p + <?>', the difference between its value and that of 

 the sum of (13) and (14) is found to be 



which, being the half the difference between the two equal second members of 

 (7) § 57 for the case of 



V 2 p = and y 2 p'=0., 



is equal to zero. Hence, the equality of the second members of (7) § 57, con- 

 stitutes the analytical reconciliation of the equations of energy for different modes 

 of generation of the same fluid motion. 



