57-58] SURFACE-DISTRIBUTIONS. 6*7 



to the first and second regions respectively, so that d/dri = - d/dn. 

 By addition, we have 



, = _ I [fl(& + W 



4t7rJJr \dn dn 



_ 



' dn 



The function <' will be determined by the surface-values of </>' 

 or d<f>'/dn' } which are as yet at our disposal. 



Let us in the first place make <'= <j>. The tangential velocities 

 on the two sides of the boundary are then continuous, but the 

 normal velocities are discontinuous. To assist the ideas, we may 

 imagine a fluid to fill infinite space, and to be divided into two 

 portions by an infinitely thin vacuous sheet within which an 

 impulsive pressure p$ is applied, so as to generate the given 

 motion from rest. The last term of (10) disappears, so that 



that is, the motion (on either side) is that due to a surface-distri- 

 bution of simple sources, of density 



4i7r \dn dn J 



Secondly, we may suppose that d<f>'/dn = d$/dn. This gives 

 continuous normal velocity, but discontinuous tangential velocity, 

 over the original boundary. The motion may in this case be 

 imagined to be generated by giving the prescribed normal velocity 

 - d(f>/dn to every point of an infinitely thin membrane coincident in 

 position with the boundary. The first term of (10) now vanishes, 

 and we have 



shewing that the motion on either side may be conceived as due 

 to a surface-distribution of double sources, with density 



It is obvious that cyclic irrotational motion of a liquid cannot be re- 

 produced by any arrangement of simple sources. It is easily seen, however, 

 that it may be represented by a certain distribution of double sources over 



* This investigation was first given by Green, from the point of view of Electro- 

 statics ; I.e. ante p. 50. 



52 



