68 IRROTATIONAL MOTION. [CHAP. Ill 



the boundary, together with a uniform distribution of double sources over 

 each of the barriers necessary to render the region occupied by the fluid 

 simply- j^mnected. In fact, with the same notation as in Art. 53, we find 



1 ((IJL ,N d /A ,7*. i !! d ( l \j **[[<* 



4>p= i- \ t \9-4fJ-r (-} + * I I j- ~ d<r, +7^ I I j- 



4?r J J an \rj 4rr J J an \rj 4n J J an 



where < is the single-valued velocity-potential which obtains in the modified 

 region, and <' is the velocity-potential of the acyclic motion which is generated 

 in the external space when the proper normal velocity - d$\dn is given to each 

 element dS of a membrane coincident in position with the original boundary. 



Another mode of representing the irrotational motion of a 

 liquid, whether cyclic or not, will present itself in the chapter on 

 Vortex Motion. 



