KECENT PROGRESS IN HYDRODYNAMICS. 69 



evanescent, whilst for water in contact with a gilded surface it is con- 

 siderable. 



The same problem has been investigated by Liibeck ' (1873), starting 

 from Meyer's solution of the equation for the stream function, and by 

 Rohrs - (1874), who took up the question from the point of view of pre- 

 cession, and considers the case where the axis of rotation slowly changes. 



The Ellipsoid. — The expressions for the velocities when a sphere moves 

 through a viscous fluid have suggested to Oberbeck ^ the form of the solu- 

 tion for an ellipsoid moving parallel to one of its axes. He shows that if ii 

 denote the potential of the ellipsoid at an external point, in the form given 



on page 17, and Q = 2 tt — then the velocities of the fluid, when the 



Jo l-^j . 



ellipsoid moves parallel to the axis a with velocity V, are given by 



L dy dxdij i 



r dQ , 2 f?'" 1 



f/\ 



where a = — s— with A = — = I /, X^ 



with A = —5 \ /, \^\t^ 



Qo + a^A .^J^(l-f-^)D. 



Tbe force on the ellipsoid necessary to keep up uniform motion is then 

 STT^t'ciE, where £ is the measure of the charge of electricity induced on the 

 ellipsoid when it is charged to potential Qq. 



The oscillations of a viscous spheroid have been treated by G. H. 

 Darwin and Lamb. In the investigations of the former,'' which are devoted 

 more directly to researches on the past history of the earth, the motions 

 are treated as very slow, and the coefficient of viscosity as large, so that 

 the problems considered belong more to the domain of elastic solids than 

 to that of hydrodynamics. The latter ^ has considered the general solution 

 of the equations of motion when the velocities are so small that their 

 squares can be neglected. The first part of his paper is devoted to the 

 solution of the system of equations 



(y2 _ 7i2) u= 0, (v2 _ /,2) ^ = o,(v' - Ji^)io = 0, and 

 dujdx + dv/dy + dwjdz = 0. 



This is then applied to investigate the oscillations of a sphere slightly 

 distorted and moving under its own gravitation, and an equation is 

 obtained on whose roots depend the time of oscillation and the logarithmic 

 decrement of the amplitude. This equation is easily solved either for the 



' ' Ueber den Einfluss, welchen auf die Bewegung eines Pendels mit einem kugel- 

 formigen Hohlraume eine in ihm enthaltene reibende Flussigkeit ausubt,' Borch. 

 Ixxvii. p. 1. 



2 ' Spherical and cylindric motion in viscous fluid,' Proc. Lond. Math. Soc, v. p. 

 125. 



' ' Ueber stationare Fliissigkeitsbewegungen,' &c. Borch. Ixxxi. p. 62. 



* • Bodily tides of viscous spheroids,' Phil Trans., part i. 1879. 



'Problems connected with tides of a viscous spheroid,' Phil. Trans, part ii. 1879. 



5 ' On the oscillations of a viscous spheroid,' Proc. Lond. Math. Soc. xiii. p. 51, 

 1881. 



