254 SIR WILLIAM THOMSON ON VORTEX MOTION. 



of the inner surface of the containing vessel, will thus be temporarily extended to 

 include each side of each of these barriers. Let now, as in § 3, any possible 

 motion be arbitrarily given to the bounding surface. The liquid is consequently 

 set in motion, purely through fluid pressure ; and the motion is [§§ 10-15, or 60, 59] 

 throughout irrotational. Hence irrotational motion fulfilling the prescribed sur- 

 face conditions is possible, and the actual motion is, of course (as the solution of 

 every real problem is), unambiguous. But from this bare physical principle we 

 could not even suspect, what the following simple application of Green's equation 

 proves, that the surface normal velocity at any instant determines the interior 

 motion irrespectively of the previous history of the motion from rest. 



61. (c). Determinacy of irrotational motion in simply continuous space. In § 57 

 (1), which is immediately applicable, as the volume is now simply continuous, 

 make q> = <p, and put y 2 <p = 0, so that <p may be the velocity potential of an 

 incompressible fluid. That double equation becomes the following single equa- 

 tion — 



where the surface integration ffdcr must now include each side of each of the 



barrier surfaces (3 V /3 2 Hence, if £<p = for every point of the bounding 



surface, we must have 



which requires that 



dx ' dy ' dz 



that is to say, if there is no motion of the boundary surface in the direction of the 

 normal, there can be no motion of the irrotational species in the interior ; whence 

 it follows that there cannot be two different internal irrotational motions with 

 the same surface normal component velocities. Thus, as a particular case, 

 beginning with a fluid at rest, let its boundary be set in motion ; and brought 

 again to rest at any instant, after having been changed in shape to any extent, 

 through any series of motions. The whole liquid comes to rest at that instant. 



A demonstration of this important theorem, which differs essentially from the 

 preceding, and includes what the preceding does not include, a purely analytical 

 proof of the possibility of irrotational motion throughout the fluid, fulfilling the 

 arbitrary surface-condition specified above, was first published in Thomson and 

 Tait's "Natural Philosophy," § 317 (3), and is to be given below, with some 

 variation and extension. In the meantime, however, we satisfy ourselves as to 

 the possibility of irrotational motions fulfilling the various surface- conditions with 

 which we are concerned, because the surface motions are possible and require 

 the fluid to move, and [§§ 10-15, or § 59] because the fluid cannot acquire 



