53-54] EXTENSION OF GREEN S THEOREM. 61 



If (/&amp;gt; also be a cyclic function, having the cyclic constants 

 AC/, #3 , &c., then Art. 43 (6) becomes in the same way 



++ 



dx dy dy dz dz 



(2). 



Eqiiations (1) and (2) together constitute Lord Kelvin s extension 

 of Green s theorem. 



54. Tf 0, $ are both velocity-potentials of a liquid, we have 



V*(t&amp;gt; = 0, V 2 &amp;lt; =:0 ..................... (3), 



and therefore 



To obtain a physical interpretation of this theorem it is 

 necessary to explain in the first place a method, imagined by Lord 

 Kelvin, of generating any given cyclic irrotational motion of a liquid 

 in a multiply-connected space. 



Let us suppose the fluid to be enclosed in a perfectly smooth 

 and flexible membrane occupying the position of the boundary. 

 Further, let n barriers be drawn, as in Art. 48, so as to convert the 

 region into a simply-connected one, and let their places be occupied 

 by similar membranes, infinitely thin, and destitute of inertia. The 

 fluid being initially at rest, let each element of the first-mentioned 

 membrane be suddenly moved inwards with the given (positive or 

 negative) normal velocity d&amp;lt;f&amp;gt;/dn, whilst uniform impulsive pres 

 sures Kip, K 2 p,...K n p are simultaneously applied to the negative 

 sides of the respective barrier-membranes. The motion generated 

 will be characterized by the following properties. It will be 

 irrotational, being generated from rest; the normal velocity at 

 every point of the original boundary will have the prescribed 

 value ; the values of the impulsive pressure at two adjacent points 

 on opposite sides of a membrane will differ by the corresponding 



