36 Proceedings of the Royal Irish Academy. 



Of the terms in M and JF each separately yields zero. 



When no term of higher order than that in M occurs in the expansion of 

 <f> the motion is of the kind which is commonly described by the statement 

 that the liquid is at rest at infinity. For such motions we see that equations 

 (2) and (3) hold good. It is only when there is uniform flow at infinity that 

 these formulae require modification. 



7. Determination of <(>. — Formula (3), modified if necessary for flow at 

 infinity, yields an implicit determination of <£, unique to the extent set out 

 in the theorems of § 3. The hydrodynamical problem is thus thrown back on 

 that of converting this implicit determination into an explicit one, in fact on 

 the problem of finding the strength of a double-sheet in the surface S which 

 shall produce a given potential field in the irrelevant region. 



In passing one wonders whether the problem of finding this surface- 

 function <£, exactly or approximately, might not be more hopefully attacked 

 by a study of the geometry of the surface and of functions and integrals 

 associated with it than by a quest for a function of position in space which 

 shall satisfy Laplace's equation and other conditions. 



8. Continuity. — It must be noted that the data of the problem are not 

 entirely arbitrary, since continuity of liquid flow requires that a certain 

 equality be satisfied by the strengths of the sources, namely 



I 



WdS = St» (4) 



If the relevant region extend to infinity in all directions the condition is 

 slightly different, being in fact that the outward flow across an outer surface *S" 

 large enough to enclose S and all the sources, together with the outward 

 motion following the motion of the boundary S, shall equal the output of all 

 the sources ; this gives 



f WdS + \d<f>/cn dS' = 2m. 



S' may be taken to be the sphere R, and in the <S" integral we may put 

 for <f> the terms set out in § 6. It then appears that the only term which 

 contributes to the integral is JW/iirB, which yields - M. So our continuity 

 condition becomes 



j WdS = M + Sot. (5) 



If M is among the data it must comply with this condition ; if it is among 

 the quaesita this condition serves to determine it. 



9. It is of course clear that if S extends to infinity, so that the relevant 

 region extends to infinity but not in all directions, the preceding results 

 may require considerable modification. But there is no real difficulty in 

 dealing with any particular case. 



