MOTION OF A SOLID THROUGH A PERFECT FLUID. 



III. On the Failure of certain Mathematical Solutions of 

 the Problem of the Motion of a Solid through a Perfect 

 Fluid. By R. F. Gwyther, M.A. 



Read October 1 8th, 1 8 8 1 . 



Of the solutions with which I intend to deal, we may take 

 that by Stokes for the motion of a sphere as the type. In 

 his paper (Camb. Trans, vol. viii. and reprint) Stokes 

 considers to what degree his solution differs from the re- 

 sults of experience, and discusses the origin of this diver- 

 gence. I propose to show that the origin is possibly of a 

 different nature to any there discussed. Stokes's solution 

 may be stated thus : If A be the centre of the sphere, of 

 radius a, AX its direction of motion at time t, (r, 6) the 

 zonal coordinates of any point in the fluid referred to AX 

 as axis, and if <j> be the velocity-potential of the motion, 



Ya z 

 <£ = const — \ —£- cos 6, 



and 



p— P T « 3 fiTV /i.VVr, za x a z . in , i , . 



^V = ^w co ^ + "^{ (3cos ^" i) "^ (3cos ^ +i } (i) 



giving the fluid-pressure. 



To deduce from this the impulsive pressures due to an 

 impulsive change in the velocity of the sphere, we write t 

 for the short time of action of the impulse, multiply equa- 

 tion (i) by dt, and integrate from t=o to t = r. Let V 



