Leathem — On Doublet Distributions in Potential Theory. 47 



Hence JT(a)->- X (a) ± 2ira l, where X<>(a) corresponds to a circular cavity. 

 Therefore 



V=X{a)+ Y{$) + Z[y), 



->- X (a) + Z"o(j3) + Z (y) i 27r(a„Z + /3<.m + 70^). 



ut JT (o) + ro(/8) + Z (y) = (V ), where (V R ) is the value of Vc, for a 

 circular cavity ; and a„l + j3 wi + y n. = 0. Hence . 



F-MFo) (34) 



This holds, on whichever side of S the point P is taken. 



Thus it appears that, while V at a point in the surface is only conditionally 

 definite inasmuch as it is represented hy a semi-convergent integral, yet the 

 limit of the value of V for a point approaching the surface is definite and is 

 the same for approach from either side of the surface, being in fact the value 

 of the above-mentioned semi-convergent integral for a vanishing circular 

 cavity. 



It will be noticed that these properties of the potential due to a surface 

 distribution of tangential doublet correspond to the properties of the tangential 

 force due to an unpolarised surface distribution of definite surface density. 1 



24. We may get further information by employing for our surface integrals 

 a method of integration by parts analogous to that commonly employed in 

 the study of the field of a solid magnet, 



Consider an area of the surface <S bounded by a curve s, and let us suppose 

 it possible to choose a set of curvilinear orthogonal coordinates (p, q) in the 

 surface S such that in the area enclosed by s both p and q are one-valued 

 functions of position, there being no curves of the families p = const., q = const, 

 which are closed curves lying wholly in the area enclosed by s. Let the 

 element of arc ds be given by 



ds 2 = P 2 df- + Qhkf, 



and let (k, fx) be the cosines of the angles which the outward normal to s at 

 any point makes with the normals to the curves p = const., q = const., through 

 the same point, all these normals being drawn in the tangent plane to the 

 surface. If ds be measured round the boundary in the sense corresponding 

 to rotation through a right angle from the direction of p increasing to the 

 direction of q increasing, 



X = Qdq/ds, fj- = - Pdp/ds. 



Let /, g be any functions of position on the surface which have definite 

 differential coefficients with respect to p and q at all points of S and s, and in 



1 Leathern, I.e. § 32. 



K.I.A. PEOC, VOL. XXXII, SECT. A. [7] 



