Leathkm — On Doublet Distributions in Potential Theory* 33 



its potential, etc., by the suffix ( ). We shall take the standard direction of 

 a normal to <S as from the relevant into the irrelevant region, and we shall 

 treat the strength r of a double-sheet in S as positive when the axis of the 

 doublet element from negative to positive points towards the irrelevant region. 



5. The Double-Sheet Potential in Hydrodynamics. — The problem of deter- 

 mining for a given region that irrotational liquid motion which corresponds 

 to a prescribed motion of the boundary is usually attacked by a search for a 

 function </>, the velocity potential, which satisfies Laplace's Equation A<f> = 

 at all points in the region (save where there are prescribed singularities), 

 and has at the boundary a prescribed normal gradient. Alternatively, 

 however, the specification of the motion may be regarded as depending on 

 the theoretically (if not practically) more simple problem of determining the 

 surface value <£ of the velocity potential. A knowledge of <£ alone would 

 give the dynamically most important function of the motion, namely the 

 kinetic energy ; and it is, in any case, easy to deduce from a known <f> the 

 general form of <j>. 



Let the boundary consist of a surface S which divides space into two 

 regions ; of these regions one is occupied by liquid, and we shall call it the 

 relevant region ; the other, though important mathematically, is not physically 

 significant, and will be called the irrelevant region. 



Let W be the normal velocity of the boundary, a prescribed function, 

 reckoned as positive when towards the irrelevant region. Let it be supposed 

 that there are sources in the liquid, the strength m of a source being measured 

 by the total time-rate of outflow across a small sphere surrounding it, so that 

 the velocity potential at small distance £ has its most important part of the 

 form - m/iwi. There may also be equal and opposite sources combined as 

 doublets of moment typified by /n. 



For purposes of integration let each source or doublet be surrounded by a 

 small sphere of radius i, and let da be an element of area on the surface of 

 such a sphere ; let dn be an element of inward drawn normal, so that 

 dn = - ds. 



Take any point P in the irrelevant region, and let r denote distance 

 measured from P. Apply Green's Theorem 1 to the part of the relevant region 

 outside the •£ spheres, using the functions <j> and r\ This gives 



r on J r on 



1 , , 

 - A© do 



i 1 r 



= ' . *k§M + *\+k§*-\**® dV ' (1) 



1 Leathern, Volume and Surface Integrals used in Physics, Cambridge Mathematical 

 Tracts, No. 1, § 18. 



[5*] 



