50 Proceedings of the Royal Irish Academy. 



p is the radius of curvature of the normal section of the surface. Thus the 



normal attraction of ds at O is of the order of Idszr' 3 or Idsp' l r~ l , so that 



the normal attraction of the whole contour at may be of the order of Ip- 1 , 



and therefore does not tend to vanishing merely on account of the particular 



power of r which it involves. This suggests semi-convergence. 



Taking 6 to be the angle which r makes with one of the principal planes 



of curvature, and (A, B) as the components of / in and perpendicular to this 



plane, 



p-' = pc 1 cos 2 6 + p 2 ~ l sin 2 8, 



where pi and p 2 are principal radii of curvature ; thus 



w = ±\(A\ + Bp) r> (pf 1 cos 2 + P p sin 2 0) ds, (41) 



a sufficient approximation being got by taking the integral round the 

 projection of the contour on the tangent plane. If this projection be a circle 

 with centre at 0, X = - cos 6, p = - sin 9, ds = rdd, and w = 0. But for other 

 forms of contour the value and limit of w may well be different from zero. 



27. A tangential component X of attraction at satisfies the equation 



X{A,B) = X(» + g/, 



where w' is the attraction-component at due to the usual line-density. We 



know that X{a) tends to a limit which depends on the shape of the vanishing 



cavity, say Xfa) ; and so 



X (A, B) = X„(<r) + Lini w', (41) 



where w' = f (AX + Bp) r' 2 cos 6 ds. 



The order in /• of the subject of integration indicates that in general a/ tends 

 to infinity as the cavity tends to vanishing, though in certain cases symmetry of 

 the cavity may make <•/ of a lower order in r _1 than appears from the general 

 formula. In such cases &»' may tend to a definite limit value, in the calcula- 

 tion of which, however, it would be necessary to take account of the difference 

 between the values of A and B at and at ds. 



28. It is, of course, clear that if div (A, B) = the tangential doublet 

 distribution in a closed surface produces zero potential and zero force at all 

 points not in the surface. At points in the surface, however, the above 

 reasoning shows that the effect of such a doublet distribution is not 

 necessarily null. 



