68 REPORT— 1882. 



Oberbeck,' and later and independently by Craig.'^ As is known the velo- 

 cities are in this method expressed by functions P, L, M, N, so that 

 u = clVjdx + cmifly - cZM/c?2 with v2P = 0. (Hjjdx + dMjchj + d'N/dz = o, 

 and the vortex rotations given by ^ = V^L, &c. He takes the most general 

 form for P in spherical harmonics with no normal velocity at the surface 

 of the sphere ; in other words, if Yn be any solid harmonic of degree n, 

 then 



'^^i^-^i^TV- 



By means of an important theorem of Borchardt's, L, M, N are then 

 expressed in the form h^= z dFJdy — y dF/dz, with similar expressions 

 for M, N, where 



The above theory applied to a sphere moving uniformly gives values 

 for the velocities which agree with those of Stokes. Expressions are also 

 given for the vortex rotations, from which it can be easily proved that the 

 vortices of equal strength lie on concentric spheres, and that the strengths 

 are inversely as the squares of the radii of the spheres on which they lie. 

 The theory given by Craig is almost identically the same, as indeed any 

 theory, starting from the basis of Borchardt's theorem and the quaternion 

 potential, must be. 



The motion of fluid inside a sphere was first investigated by Helm- 

 holtz,-' allowance being made for a slipping of the fluid over the surface 

 of the boundary. The paper in which his investigation appeared was 

 a joint one, containing the mathematical theory by Helmholtz, and the 

 experimental by Piotrowski. The motion is considered so small that 

 squares of the velocities may be neglected, and in this case the motion is 

 such that it may be represented by supposing concentric shells of the 

 fluid to revolve as if rigid with an angular velocity depending on the 

 radius of the shell, the most general expression for which is a sum of 

 terms of the form 



w„= Ae"* < -^ cosli /3r sinh (Sr > where /3 = ± >/(o//i'), 



and a is in general a complex. The motion can be represented as a series 

 of waves propagated to the centre with rapidly decreasing intensity, and 

 there reflected. If, for example, the boundary have a simple har- 

 monic motion of period -, the velocity of propagation of these waves 

 will be 2^ (^iTr/r), and is therefore dependent on the time of vibration. 

 Whilst the wave moves through a wave-length 2\/(/i7rr), the amplitude 

 diminishes in the ratio 1 to exp.( — 27r), or from 1 to 1/535. Helmholtz 

 works out the motion for the sphere as applicable to the data in 

 Piotrowski's experiments. These experiments contain the first attempt 

 to approximate to the value of the coefficient of gliding. The values 

 of this found for alcohol and ether are so small that they are probably 



' ' Ueber stationiire Fliissigkeitsbewegungen mit Beriicksichtigung der inneren 

 Eeibung,' Bvrch. Ixxxi. p. 62, 1875. 



2 ' On steady motion in an incompressible viscous fluid,' Phil. Mag. (.5) x. p. 342 

 (1880) 



' ' Ueber Eeibung tropfbaver Fliissigkeiten,' Sitzber. d. h. Akad. ^S'iss. Mien., si. 

 p. 607 (1860), and WissemchaftUchen Abhandlungen, i. p. 172. 



