9Y 



A New Problem in Hydrodynamics with Extraneous 

 Forces Acting. 



By Edwakd Lee Haxcock. 



The solution of most problems in liydrodynamics depends upon ilm 

 proper coml)ination of the equations of motion of the fluid interior of 

 a given closed surface with the difi"(>renlial eciuatiuii of the surface, or 

 with tlie e(iuati(ins expressing the Loundary conditions. 



Lord Kelvin has shown that the differential equation of the surface 

 for )ioth compressilile and incompressible fluids has the following form: 



u.F'(x) + v.F'(y) -f w.F'(z; + F'(t) = 

 wliere (t) is a variable parameter of the equation 



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



In the ti'catment of problems of the motion of incompressible fluids 

 in three dimensions, where the surface under discussion is spherical 

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

 ( ^72 -_ )^ 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 

 symmetry of the boundary conditions. Where the surface differs much 

 from the splierical form as in ellipsoids, ellipsoidal harmonics are used. 

 Problems of this kind have been extensively investigated. 



In discussing the anchor ring Mr. "S^'. M. Ilick.s^ has derived modified 

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

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

 tigated. The same problem has been solved by Mr. F. W. Dyson- hj 

 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. Phib Trans. ]893. 



2. Phib Trans. 1881, Part III. 



7— A. OK SciENcr, '03. 



