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 



<l> = m/r (1), 



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

 a radial flow from the point, and if SS, = r 2 8o7, be an element of a 

 spherical surface having its centre at the surface, we have 



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 ra', at a 

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

 small, and ra' infinitely great, but so that the product ra'Ss is finite 

 and equal to fi (say), is called a ' double source ' of strength //,, and 

 the line Ss, considered as drawn in the direction from ra' to + ra', 

 is called its axis. 



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

 double source ^ situate at (of, y', z'\ and having its axis in the 

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

 f(af + IBs, y' + mSs, z' + nSs) -f(x', y', z') 



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



r = {(x - xj + (y - yj + (z - 



we find = M + m + n / ............... ( 2 ), 



dx dy dz J r 



d , d d\l 



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



dx dy dz) r 



cosS- 



where, in the latter form, ^ denotes the angle which the line r, 

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

 axis (I, ra, 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. 



