Table of Zonal Spherical Harmonics. 523 



ingenious quick way of working ; but we had to be taught by 

 experience. 



Example III. 



A solid bounded by a surface of revolution moves axially in 

 an infinite mass of incompressible fluid which has no other 

 motion than this gives to it. Find the motion. 



In this case 



V 2 V = 0: 



dV 

 and — -y- at any point at the surface of the solid (dn being 



an element of the normal) in the fluid is really the normal 



velocity of the solid itself. Again, V must be constant at any 



infinite distance. 



This may easily be worked out for any surface of revolution. 



Applying it to a sphere of 1 centim. radius. Let v be the 



velocity of the centre of the sphere in the direction parallel to 



dY 

 the axis of z. Then -=- = v cos 6 at any point at the surface 



of the sphere. 



Expressing V in zonal harmonics, and taking it as at an 



infinite distance, 



V= ^ + ^ + ^ 2 + &c. 



r r" r 



dV , 1 . 



-r- wnen r=l is 

 dr 



V cos6=- A P - 2 AiPi - 3 A 2 P 2 - &c. 

 But P ] =/jl= cos 0, so that the other coefficients vanish, and 



v 



A ="2< 



so that 



v^-ijp, (1) 



The equipotential surfaces which a student will draw from (1) 

 are perfectly well known. It is a good exercise to draw them. 

 If now a velocity — v is impressed upon the whole system, 

 sphere and fluid, we have the case of the sphere at rest and 

 the fluid moving past it. We now merely add the term 

 -wcos^ to (1) and obtain 



v — v { r+ w)^' co 



which again is easily represented upon paper in equipotential 

 surfaces, 



