Leathem — On Doublet Distributions in Potential Theory. 49 



From this it follows that the limits of the potential and of the force 

 components, whether normal or tangential, at P, for P -> 0, where is a 

 point in the surface, are the same as in the well-known case of an ordinary 

 surface density, so that no special discussion of these is called for. 



■ 25. Potential and force at a point in the surface. — For a point 0, however, 

 which is situated in the surface S, the equivalence of the tangential doublet 

 distribution and the surface density - div (A, B) does not hold good. For 

 must be surrounded by a vanishing cavity, and the limit of the line integral 

 round the edge of the cavity is not necessarily negligible. 



Thus if we represent the potential at due to the surface density 

 a = - div (A, B) by V (<j), which potential we know to be independent 

 of the mode of vanishing of the cavity, and if V (A, B) stand for the 

 potential at of the tangential doublet distribution, which we know is 

 not independent of the mode of vanishing of the cavity, we have 



V (A, B) = V (a) + Lim w (38) 



where w = f ;■"' (XA + fiB) ds, (39) 



the line integral being taken round the edge of the cavity, and it being 

 remembered that (A, ,u) correspond to the normal drawn inwards to the 

 cavity. 



It is readily seen that if the cavity be of any form which is symmetrical 

 about the line through perpendicular to the direction of the resultant / 

 of (A, B), w -> 0. For cavities of this class Vo (A, B) = V (a-). 



But if, for example, we take a cavity which vanishes in the form of a 

 rectangle with sides parallel and perpendicular to /, the former tending to 

 infinite smallness in comparison with the latter, and being on the shorter 

 central line and dividing it in the definite ratio 8 : 1, it is easily calculated 

 that w ->■ ± 21 log 8. So Lim w does not vanish for all cavities, and therefore 

 there is not complete equivalence as regards potential between the / distribu- 

 tion and the <r distribution. 



26. Passing to the consideration of the normal attraction A T at O, we note 

 that when there is a cavity round 



N{A,B) = N(a) +a, 



where id is the normal attraction at due to the line density \A + fiB in 

 the contour ; and we remember that N (a) tends to a definite limit JV (<r) as 

 the contour closes in round in any manner. Thus 



iV (A, B) = N (<r) + Lim w. (40) 



If is a point at which the surface S is free from singularity, and if z be 

 the distance of ds from the tangent plane at 0, z is approximately h'~p~ l where 



