52 On Stokes s Current Function. [Dec. 11, 



The applications to hydrodynamics whirh I here give are of 

 mathematical interest rather than physical. They are chiefly in con- 

 nexion with the motion of viscous liquids. In ' Crelle-Borchardt,' 

 vol. 81, 1876, Oberbeckhas given the velocities produced in an infinite 

 viscous liquid by the steady motion of an ellipsoid through it, in the 

 direction of one of its axes, and from these Mr. Herman* has found 

 the equation of a family of surfaces containing the stream lines 

 relative to the ellipsoid. In chapter vi, Stokes's current function 

 is obtained by a direct process for the flux of a viscous liquid past a 

 spheroid, and it is shown that the result differs only by a constant 

 multiple from the particular case of Mr. Herman's integral. 



Some minor applications are also given, namely, the solutions are 

 obtained for flux past an approximate sphere, and past an approxi- 

 mate spheroid. The solution is also obtained for flux through a 

 hyperboloid of one sheet, where it appears that the stream surfaces 

 are hyperboloids of the confocal system. A particular case is that 

 of flux through a circular hole in a wall, and this is interesting 

 because we see that, by supposing internal friction to take place in 

 the liquid, we find an expression which gives zero velocity at the 

 sharp edge, and thus avoids the difficulty which is always present 

 in the solution of such problems on the supposition that the liquid 

 is perfect. A comparison may be instituted between this problem, 

 and that of the effect of a disturbing periodic force upon a 

 dynamical system capable of vibrating alone with a period eqnal 

 to that of the force. It is well known that the amplitude of the 

 vibration induced appears infinite, if we totally disregard friction, 

 and this difficulty is met by the fact that the damping effect of even 

 slight friction is rendered considerable by high velocities. Now a 

 viscous liquid can move irrotationally, and, if there were no friction 

 at the boundaries, this is the class of motion it would take in cases 

 of flux past or through obstacles. But if the obstacle terminated in 

 a sharp edge, this would make the velocity there infinite, and the 

 friction, however inconsiderable elsewhere, would here become of 

 account. The boundary conditions which were necessary for the 

 existence of irrotational motion throughout the liquid would no 

 longer apply, and the whole character of the solution would be 

 changed. This would at any rate seem to apply to cases in which 

 the whole motion is slow, and when, consequently, the boundary con- 

 ditions which must hold are pretty well understood. 



The paper concludes with an attempt to discuss the flux past a 

 spheroid, or through a hyperboloid at whose boundary there may be 

 slipping. The current function is not obtained, all that appears 

 being that it probably differs from the parallel case of the sphere in 

 being far more complicated than when there is no slipping. From 

 * Quart. Joura. Math.,' 1889 (No. 92). 



