38 Proceedings of the Royal Irish Academy. 



So the final result is 



\AdS = \w^aS-Vmi-vJL^+M^ + N d 4-X (8) 

 J dn J \ dx oy dz I 



12. It is known 1 that the potential of a double-sheet of strength rata 

 point in the sheet is definite, and that it differs by + 2ttt from the limit 

 of the potential at a point which moves up to the sheet from either side. 

 Consequently if we let the point P from which the /■ of formula (2) is 

 measured move up to the surface S we get the limit formula 



the surface integrals being known to be convergent. 



Combining this with (8), assuming \p to have no singularity on the surface 

 S, we get 



S^,+j*^@++)d»-jV(l + *)<IS-S.(l + +) 



-»H**h*'s$**)- <10 > 



In the absence of sources and doublets we can deduce a formula for the 

 kinetic energy T, namely, 



2T= \$'W'dS'= ~^ww(^+ 4?\dSd&~ [w$^(± + J\dSdS', 



(11) 

 these being integrals taken twice over the surface S. 



It is conceivable that, for a particular form of S, some happy choice 

 of -^ might make it possible to evaluate the integrals in (10) or (11) either 

 accurately or approximately. 



13. Approximate form of <f> at great distance. — Returning to the terms set 

 out in § 6 as possibly representing the most important part of the velocity 

 potential at great distance from S, and thinking in particular of the motion 

 in liquid extending to infinity due solely to the motion in it of a rigid body 

 whose surface is the boundary S, we see that the approximate form of $ at 

 great distance B from a definite origin is - J/' cos d'/iirHr, or 



- (AL + BM + CN)I±ttR\ (12) 



where L. M. Nate the direction cosines of B. This shows that the motion is 

 to this approximation the same as would be due to a doublet at the origin 



! Leathern, I.e., §34. 



