118 Mr. A. L. Selby on two 



The doublets BP reduce to a source fib/c at P and a line- 

 sink ^p along BP. 



The doublets QC reduce to a source fjbabj(c 2 — b 2 ) at Q and 

 a line-sink — ~' 9 — ^ along QC. 



We shall find that the doublets BD contribute a term of 

 order c -10 to T ; this we shall neglect. 



The lengths of successive line- sinks diminish very rapidly, 

 each being to its predecessor in a ratio of the order cf/c 2 or 



Therefore, since B~P = b 2 /c, the lengths QC, ED, . . . are of 

 order c -3 , c~ 5 . . . , and the strengths of successive sources 

 A, P, Q, . . . are of order 1, c -1 , c~ 2 . . . 



If a source v at distance r from A contribute a term (j> v 

 to <fi A , it has been proved by Gauss that 



J J</) i/ JS 1 = —A.ira?vlr or —Airav, according as r> or <a. (5) 



[This may be proved in the following elementary way : — 

 Let the sphere be an attracting shell of surface-density 

 unity. 



f §<fi v d$ is the work done by a particle v if the sphere recedes 



to infinity. 



And— 4z7ra 2 vjr or — 47ravis the work done by the sphere 

 when the particle recedes to infinity, according as r > or < a.] 



Excepting the source /jl at A, all the images within A are 

 of total strength zero. 



An image outside A consists of a source of strength s 

 distant p from A, and a line-sink — s/y per unit-length, with 

 extremities distant p, p + y from A. 



This contributes to the coefficient of 47ra 2 in — C J* A c?S x 

 the term 



£ fvs dx 



P Jo 7 P + x' 

 or 



which is, approximately, 



sy/2p 2 - S y*/3p 3 (7) 



For the image PB, 



8 = fib/c ; y = b 2 /c ; p = c-b 2 Jc. ... (8) 







