Pressure around Sj^Jieres in a Viscous Fluid. 433 

 From (15) "deb «' 



^^ = -i^%-^^n^, (21) 



^y = — ^'cos6', (22) 



and "^0 _ i^sin ^ ^^^on 



§7 "^ r ^ ^ 



From (16) the pressure aromid a single sphere moving in 

 a perfect fluid is 



^=|cos-^^-f (24) 



The broken-line curve nn in Plate XVIII. tig. 9 exhibits 

 the pressure of a perfect fluid around a single sphere when 

 moving with constant velocity. 



In order to obtain the pressure-distribution around two 

 spheres in a perfect fluid w^e determine the velocity potential 

 and solve equation (16). The velocity potential for two 

 spheres moving in their line of centres may be obtained ap- 

 proximately by the theory of images*. Using only the 



terms in the expansion of the first image so far as ( - j we 



obtain for the velocity potential of two spheres moving in 

 their line of centres, fig. 7, 



-.cos ^4-^ V — (c—rcosU) { 



H ^ C \ G 



cf)z=^u~cos + ^ V -^ {c—r COS 6) -J 1 + 3 — cos 6 



,, r- 5 cos'^ — 1 ^T^ 1 cos'^^ — 3 cos 6 

 + 3- ^ +0-3 — 



C- "L C 



where a and h are the radii of the spheres, r the distance from 

 the centre of sphere A, c the distance apart of the two spheres, 

 and u and v are the respective velocities. For constant 

 velocity u = v and at the surface of sphere A, r = h~a and 



^^ /I 1 «^ ^ r^ o<^^ /I , o^'^cos""^— 1 



^--= -z^cosa' — iit-cos^S 1 + 3 — cos ^ + 3 — r, 



or 2 ^,3 L c c- 2 



.a'7cos'6>-3cos^^ . d /c zj \ / o t* n 



+ 5^-, j^i,_(^-_cos^j|3-cos^ 



_^^.^5_co^l_^^.^^7^^^ (26) 



* Stokes, /. c. p. 1 ; Hicks, I c. p. 2. 

 Phil. Mag. S. 6. Vol. 6. No. 34. Oct. 1903. 2 F 



