97 



A New Problem in Hydrodynamics with Extraneous 

 Forces Acting. 



By Edward Lee Haxcock. 



The solution of most problems in hydrodynamics depends upon thu 

 proper combination of the equations of motion of the tluid interior of 

 a given closed surface with the differential eciuation of the surface, or 

 with tlie (Hiualimis expressing the boundary conditions. 



Lord Kelvin has shown that the differential equation of the surface 

 for botli comprcssilile and incompressilile fluids has the following form: 



u.F'(x) + v.F'(y) + w.F'(zj f F'(t) = 

 where (t) is a varialile parameter of the equation 



F (X, y, z, t)=0. 



In the treatment of problems of the motion of incompressible fluids 

 in tlH'ee dimensions, where the surface under discussion is spherical 

 or nearly so. the usual particular solutions of Laplace's equation 

 ( ^2 ^ __ (J j_ such as, zonal, tesseral and spherical harmonics, are 

 adequate, since in these cases the velocity-potential satisfies Laplace's 

 equation. The solution used in any particular case depends upon the 

 symmeti-y of the boundary conditions. Where the surface differs much 

 from the .'-pherical form as in ellipsoids, eUipsoidal harmonics are used. 

 Problems of this kind have been extensively investigated. 



In discussing the anclior ring :Mr. V>' . M. Hides' has derived modified 

 forms of the zonal, tesseral and spherical harmonics by means of which 

 the potential both outside and inside the ring may be completely inves- 

 tigated. The same problem has been solved by ^Nlr. F. W. Dyson- by 

 using elliptic integrals. 



The problem is much simplified when the motion takes place in a 

 single plane, in which case, if the boundary consists of a straight line, 

 two parallel straight lines, or is rectangular, the velocity-potential may 

 be expressed as a Fourier's series or a Fourier's integral. 



1. Phil. Trans. 1893. 



2. PhiL Trans. 1881, Part III. 



7— A. OF SciEXCK, '03. 



