64 IRROTATIONAL MOTION. [CHAP. Ill 



The velocity-potential at any point P, due to a simple source, 

 in a liquid at rest at infinity, is 



$ = m/r (1), 



where r denotes the distance of P from the source. For this gives 

 a radial flow from the point, and if 8S, = r^vr, be an element of a 

 spherical surface having its centre at the surface, we have 



-^- dS = 47nft, 

 dr 



a constant, so that the equation of continuity is satisfied, and the 

 flux outwards has the value appropriate to the strength of the 

 source. 



A combination of two equal and opposite sources + m , at a 

 distance 8s apart, where, in the limit, 8s is taken to be infinitely 

 small, and mf infinitely great, but so that the product m 8s is finite 

 and equal to p (say), is called a double source of strength /A, and 

 the line 8s, considered as drawn in the direction from m to + m , 

 is called its axis. 



To find the velocity-potential at any point (x, y, z) due to a 

 double source ft situate at (# , y , z \ and having its axis in the 

 direction (I, m, n), we remark that, / being any continuous function, 

 f(af + 18s, y + m8s, z + n8s) -f(x, y , z ) 



ultimately. Hence, putting /(a/, y , z) = m /r, where 

 r = {(x - xj + (y- yj + (z - /) }*, 



ni /, d d d\ I 



we find &amp;lt;t&amp;gt; = ^l- + m - + n ^- ............... (2), 



d d d\l 



+ m- r +n-j- - ............... (3), 



dx dy dz) r 



where, in the latter form, S- denotes the angle which the line r, 

 considered as drawn from (x, y , z) to (x, y, z}, makes with the 

 axis (I, m, n). 



We might proceed, in a similar manner (see Art. 83), to build 

 up sources of higher degrees of complexity, but the above is 

 sufficient for our immediate purpose. 



