RECENT TROGRESS IN HYDRODYNAMICS. 47 



iuside au ellipse, and inside a semi-ellipse, by the elliptic transformation 

 from the solations for the circle obtained by Greenhill. 



The images inside an ellipse due to either a source, or a doublet 

 outside it, have been determined by the writer.' In general it may be 

 taken to consist of a line distribution of sources and doublets along the 

 straight line joining the foci, and of an isolated image or not, according 

 to the position of the original soni'co. When this lies beyond a certain 

 confocal ellipse, determined by the size of the bounding ellipse, there is 

 no isolated image, -whereas, if it lies within, there is an isolated imago 

 lying on the confocal hyperbola through the 'object.' 



Where the ellipse degenerates into the line joining the foci the 

 isolated image is always absent, and there is only a distribution of 

 doublets along the line. The densities at a point are given throughout 

 in terms of the position of the point and of the object soui'ce or doublet ; 

 these expressions for certain particular positions of the source become 

 very simple. Thus, in the case of a line AB, and a doublet at P,, on 

 ,AB produced, and perpendicular to it, the line doublet density at a point 

 P on the line is proportional to 



PiV Upi.bpJ- 



pp 



The motion of a mass of fluid in the form of an elliptic cylinder 

 rotating about its axis under the attraction of its own mass has been 

 touched upon by Dirichlet and Riemann in their investigations on the 

 similar problem for the ellipsoid referred to below. More particularly, 

 Lipschitz 2 gives equations for the motion of such an ellipse, both for 

 vibrations of form and rotation, and shows that they are purely periodic 

 between definite limits. Kirchhoff '^ has given a simple case, where the 

 boundary rotates without change of form — a case which is embraced in a 

 more general solution of the same problem given by Greenhill.^ The 

 latter considers the motion to be generated by supposing the fluid to 

 rotate within a rigid boundary as a solid body with angular velocity w, and 

 an additional angular velocity w' to be impressed on the boundary. He 

 finds that we may suppose the boundary removed, provided the relation 

 between these quantities and the axes is given by 



The paths of the particles are in general pericycloids, which, (1) 

 when w' = w (a^ _ h'^) {(a- + h^) are epicycloids, (2) when w + w' = o, or 

 boundary at rest, are ellipses (Stokes's case referred to above, p. 8), 

 (3) when w = o are circles, and (4) when w' = — w (a + hyj^a"^ + h^) 

 are circles, which last is Kirchhoff's case. 



OtJier Curves. — Any number of possible fluid motions can be deter- 

 mined by taking any solution of the equation V"^ = o» ^n<i determining 

 the stream lines, any one of which may serve as a boundary. But this 



' ' On functional images in ellipses,' Quart. Jour. xvii. p. 327 (1881.) 



- ' Reduction der Bewegung eines fiiissigen homogenen Ellipsoids aiif das 

 Variationsproblem eines einfachen Integrals, nnd Bestimmung der Bewegung fiir 

 den GrenzfaJl eines nnendlichen elliptischcn Cylinders,' Borcli. Ixxviii. p. 245 

 (1874). 



« ' Vorlesungen,' &c. p. 262. Aufi. ii. 



' ' On the rotation of a liquid ellipsoid about its mean axis,' Proc. Camb. PMl. 

 Soc. iii. p. 233. 



