Leathem — On Doublet Distributions in Potential Theory. 43 



illation can be made up by combining uniform magnification, rigid body- 

 displacement, and inversion, the last of which is the only operation involving 

 analytical difficulty in its details. It may therefore be of interest to examine 

 how doublet distributions and their fields fit into the method of inversion, 

 and incidentally to consider the illustration from hydrodynamics suggested 

 by the similarity already indicated between the double-sheet problem and the 

 problem of liquid motion. 



The method of inversion of potential fields is set out in Thomson and Tait's 

 Natural Philosophy, §§ 515, 516. A centre and radius k of inversion being- 

 chosen, r and r being the distances of a point Q and its inverse Q' from 0, 

 dl, dS, civ being elements of length, area, and volume in one configuration, and 

 oil', dS', civ corresponding elements in the inverse configuration, it is known 

 that 



ell' = r -cll, dS' = '^clS, civ' = T ~clv. (22) 



If, further, a particle of mass m placed at a point A produce a potential 

 Fat a point P, and a particle of mass ml = mr'/k placed at the point A' 

 inverse of A produce potential V at the point P 1 inverse of P, then 

 V' = Fk/B', where r, r, B, B' are the radii vectores from to A, A', P, P' 

 respectively. 



The particles typified by m and ml can be generalized into surface densities 

 a, a', and volume densities p, p', related to one another by the laws 



a = <jk 3 /r' s , p = pk 5 /r'% (23) 



and for these distributions the law of potential correspondence V = Vlc/B' 

 still holds. 



17. — It is clearly legitimate to extend the same sort of correspondence to 

 doublets, but the result is less simple. A doublet of strength p situated at 

 Q, with its axis inclined at an angle ^ to OQ, is the limit of particles 

 (- m, + m) at a distance dl apart where mdl -> p for dl ->- 0. The 

 inverses of these are particles 



— ml = - mr'/k, ml + dm' = m (;•' + dr')/k, 



at a distance dl' apart, where dl' = (» /2 /^ 2 ) dl and dl' makes an angle tt - y. 

 with OQ'. 



Now passing to the limit for dl ->- we get at Q not only the doublet 

 p = Imxm'cll' = pr' 3 /k 3 , (24) 



but also a particle of mass v where 



v = lim mdr'/k = lim (- dl' cos %Jk) {ivlk/r') = - (p,'/r') cos x 



= - (pr' 2 /k 3 ) cos x = - (pk/r*) cos x . (25) 



