53-54] EXTENSION OF GREEN'S THEOREM. 61 



If p also be a cyclic function, having the cyclic constants 

 Ki t K 2 f , &c., then Art. 43 (6) becomes in the same way 



_ HI fdp dp dp dp d$ dp\ 

 JjJ \dcc dx dy dy dz dz ) 



dy dy 



(2). 



Equations (1) and (2) together constitute Lord Kelvin's extension 

 of Green's theorem. 



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



V 2 < = 0, V 2 <' = ..................... (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<f>/dn, whilst uniform impulsive pres- 

 sures Ktf, tc 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 



