46 Proceedings of the Royal Irish Academy. 



brackets is absent., and (31) is identical with equation (2), the term - 4nrtp x 

 (if present) corresponding to the 4ttC of § 6. If is in the relevant region 

 of the original motion the first term in brackets must be retained, and (31) 

 is now identical with equation (6). 



Thus it appears that the continuity condition of the inverse motion is 

 identical with relation (3). the relation which determines <j>, or with relation (7), 

 the relation which specifies $ in terms of <j>, according as the centre of 

 inversion is taken in the irrelevant or the relevant region of the original 

 motion. This pair of alternative identities in the inversion transformation is 

 rather remarkable. 



22. Surface Distribution of Tangential Doublet. — In the most general surface 

 distribution of doublets the moment in an element of surface dS has com- 

 ponents (adS. (3dS. jd.S) where (a, /3, y) are functions of position on the 

 surface. If (2f, »;, £) are the coordinates of dS the potential at a point 

 P (x, y, z) is 



V = \{a(x - %) + P(y - n ) + , . - ; r*d8. (32) 



If (/, m, 7i) be the cosines of the normal at dS, and if at all points on S 



ol + fim + yn = 0, (33) 



then the doublet distribution is one of tangential moment. 



It is clear from the form of V that., if P be in the surface S or be made to 

 approach a limit position O in the surface, questions of convergence arise 

 with respect to the integrals representing the potential and the components 

 of attraction. 



23. Limits of potential and force for a, •point approaching the surface. — 

 Applying the standard tests 1 we consider first the potential V, and we notice 

 that the potential V at is a surface integral whose subject of integration 

 tends to infinity at like r 1 cos ft, where 6 is the angle between r and a fixed 

 direction in the surface. This points to a semi-convergence, namely convergence 

 to a value which depends on the shape of the limiting cavity round 0. In fact 

 Vc, is the sum of the x, y, z components of attraction at of ordinary surface 

 densities a, 0, y, respectively, (say X (a), T(ji , Z{y)), and it is known that 

 each of these is generally semi-convergent. 



The limit of V for P ->- is the sum of the limits of the above-named 

 attraction components at P. Xow a tangential attraction at P due to an 

 ordinary surface density tends to a limit which is the corresponding attraction 

 at for a circular cavity, but a normal attraction tends to a limit which 

 differs from the corresponding attraction at by 2s- times the surface density. 



1 Leathern, l.c-, section viii. 



