66 IRROTATIONAL MOTION. [CHAP. Ill 



potential due to a surface distribution of simple sources, with a 

 density 1/4-Tr . d(f&amp;gt;/dn per unit area, whilst the second term is the 

 velocity-potential of a distribution of double sources, with axes 

 normal to the surface, the density being 1/4-Tr . &amp;lt;. 



When the fluid extends to infinity and is at rest there, the 

 surface-integrals in (7) may, on a certain understanding, be taken 

 to refer to the internal boundary alone. To see this, we may take 

 as external boundary an infinite sphere having the point P as 

 centre. The corresponding part of the first integral in (7) 

 vanishes, whilst that of the second is equal to G, the constant 

 value to which, as we have seen in Art. 41, (f&amp;gt; tends at infinity. 

 It is convenient, for facility of statement, to suppose (7=0; 

 this is legitimate since we may always add an arbitrary con 

 stant to &amp;lt;. 



When the point P is external to the fluid, &amp;lt; is finite through 

 out the original region, and the formula (5) gives at once 



o ds + ds ......... (8), 



47rJJ r dn 4t7rJJ ^ dn\rj 



where, again, in the case of a liquid extending to infinity, and at 

 rest there, the terms due to the infinite part of the boundary may 

 be omitted. 



58. The distribution expressed by (7) can, further, be re 

 placed by one of simple sources only, or of double sources only, 

 over the boundary. 



Let &amp;lt;f&amp;gt; be the velocity-potential of the fluid occupying a certain 

 region, and let (/&amp;gt; now denote the velocity-potential of any possible 

 acyclic irrotational motion through the rest of infinite space, with 

 the condition that &amp;lt;/&amp;gt;, or &amp;lt; , as the case may be, vanishes at infinity. 

 Then, if the point P be internal to the first region, and therefore 

 external to the second, we have 



47rJJ r dn 4m ]] ^ dn 



(9), 



o , 



4?r Jj r dn 4-TrJJ ^ dn 



where $n, &n denote elements of the normal to dS, drawn inwards 



