Pulsating Spheres in a Liquid. 119 



Therefore this contributes 



ph fi , c 2 



The source at E is of order c~ 3 ; RD is of order c~ 5 ; AR is 

 of order c. 



Therefore RD contributes a term of order c -10 . 

 Omitting terms of this order, 



-«J5 <MS 1= W<?{i + j^. 2 - I log ^} ; 

 — 6 (T </> E <^$ 2 can be written down by interchanging a and 5. 

 It remains to form the term —§§ <£ A dS 2 i* 1 (4) . 



A contributes a 2 a/c to the coefficient of 47rfr 2 ; BP contri- 

 butes nothing. 



For QC, s = pbab/(c 2 — b 2 ) ; p = c — a 2 c/(c 2 — b 2 ), p being 

 here reckoned from B ; 



7 c(c 2 -b 2 ) W 



sy 2 /3p 3 is of order c~ u ; 



sy _ fia 3 b 3 /l a 2 + £ 2 \ 

 2^ " c 7 \2 + c 2 ) ; 



RD contributes nothing, and further images are negligible. 

 Therefore, to terms of order c~ 9 inclusive, 



— = 4<7raV -j - + -a — t.—j log -5 — r 2 f 

 p Ca (T — V b Q c 2 — b z J 



+ 47rW (i+ -r^-i log -^-- £ 



C6 c 2 — cr a fe c 2 — a 2 J 



Now the force tending to increase c, required to maintain 



c at its given value, is — . 



55 ' dc 



Therefore, if the spheres are free to move they behave 



dT 

 as if a force -=— tended to separate them, and (omitting for 



brevity terms beyond c~ 8 ), 



ldT_ A,ira 2 b 2 ab f^ 7a*b\ i.irda 2 b z (^ W \ _±irl>Hr \f / '2a 2 \ 

 pfa~~~ c 2 \ 2c 6 J & \ ±+ (y i ) c 6 \ l+ r)' 



