48 



Proceedings of the Royal Irish Academy. 



the following forniulge let the line integral be taken round the complete 

 boundary and the area integrals over the whole of the enclosed area ; then 



\(\f + M)ds = \{fQdq - gPdp) 



This is, of course, Stokes' Theorem, 

 the fact that 



Its presentation in this form brings out 



is independent of the choice of coordinates, being from one point of view the 

 divergence of the tangential vector represented by (/, g), say div (/, g), and 

 from another point of view the normal component of the curl of any vector 

 whose tangential part is represented by (- g, /). 



In the foi-mula (35) let us put f=Ar~\ g = Br~ l , where r is distance 

 from P, and {A, B) are the components in the directions of <p and q of the 

 density of tangential doublet-moment. We get 



XA + „B 



ds 



pq [dp 



Q' 



dq\ r 



dS 



)p (QA)4 q (PB))ld Sl 



Ad_ Bd_ 

 Pdp + Qdq 



dS, . 



(36) 



of which the last term is clearly the potential at P. Hence we have 



V = - j r' div {A, B) dS + [ r 1 {XA + fiB) ds, . . . (37) 



which shows that the tangential doublet distribution produces the same field 

 of potential as an ordinary surface density - div {A, B) and a line density 

 {XA + fiB) in the boundary edge. 



If P is not in the surface, and if the surface is a closed surface of one 

 sheet, there is no boundary edge. The formula (37) can be applied separately 

 to the two portions into which the surface is divided by a closed curve drawn 

 upon it. For these two portions the cosines (X, /u) are of opposite sign, and 

 so the two line integrals add up to zero. Thus the tangential doublet 

 distribution produces the same potential at points not in the surface as a 

 surface density - div {A, B). 



