52 IRROTATIONAL MOTION. [CHAP. Ill 



Let T be the kinetic energy of the irrotational motion to which 

 the velocity-potential cf> refers, and 2\ that of another motion 



given by 



dd> d<b , dd> 



W= - + W V = ~ + V > W = ~ 



where, in virtue of the equation of continuity, and the prescribed 

 boundary-condition, we must have 



du Q dv Q dwp _ - 

 dx dy dz 



throughout the region, and 



Iu + mv + nw = 

 over the boundary. Further let us write 



T^^ptfKuf + vf + wfidxdydz ........... .(6). 



We find 



Since the last integral vanishes, by Art. 42 (4), we have 



T l= T+T a .......................... (7), 



which proves the theorem. 



46. We shall require to know, hereafter, the form assumed by 

 the expression (4) for the kinetic energy when the fluid extends 

 to infinity and is at rest there, being limited internally by one or 

 more closed surfaces $. Let us suppose a large closed surface 2 

 described so as to enclose the whole of 8. The energy of the fluid 

 included between 8 and 2 is 



where the integration in the first term extends over S, that in the 

 second over S. Since we have by the equation of continuity 



(8) may be written 



where G may be any constant, but is here supposed to be the 

 constant value to which </> was shewn in Art. 39 to tend at an 



