46 REPORT— 1882. 



rotating rectangle are given in infinite series.' The path of a vortex 

 inside such a rectangle ^ admits of a very elegant expression by means of 

 elliptic functions ; as well as the <•/> and \p for sources and sinks at the 

 corners.^ For instance, for a source and a sink at two adjacent comers on 



the same radius <f> + \pi = log sn — { + i log - J with analogous expres- 

 sions in en and dn for the other corners. Here the value of q is (a/b)\ 

 The position of rest of a single vortex is at a distance from the centre 

 equal to the geometrical mean of the radii.* The solution for sources is 

 also given by Allen. ^ 



Ellipse (axes a, b ; a > h). If the elliptic cylinder be considered as 

 the limiting case of an ellipsoid when one axis becomes indefinitely large. 

 Green may be regarded as the first worker in this field (1833) ; but the 

 first to consider definitely the ellipse was Stokes (1843) in his before- 

 mentioned paper 'On some cases of fluid motion'; but in this he only con- 

 sidered the motion approximately, in the space outside an ellipse of small 

 eccentricity, for translation and rotation. He*' has also shown that con- 

 focal conies are possible forms for stream lines, though the motion is only 

 irrotational in the case of the rectangular hyperbola. In 1873 Beltrami^ 

 gave the velocity potential for the motion of an elliptic cylinder as a case 

 of the ellipsoid, whilst Ferrers ^ determined the stream function two years 

 later (1875) for motions of translation and rotation. In the latter year 

 also Lamb-' published the expressions for ^j and \p in the forms which 

 are now generally used, and given in KirchhofF's ' Vorlesungen ' and 

 Lamb's ' Theory of Motion.' In this paper will be found diagrams of 

 the lines of flow. The path and motion of an ellipse moving in an infinite 

 fluid have been worked out by Greenhill, '" who has given figures illustrating 

 the motion for the three cases when it is projected, so as to make (1) 

 oscillations, (2) whole revolutions, and (3) when it is projected in the 

 direction of the major axis with infinitely small angular displacement. 



The same author ' ' has also investigated the motion of an ellipse 

 whose centre is fixed in a stream. In this case the time of a small 

 oscillation is 27r ^,/ [kaah(a'^ + 1^) l(a~ — l'')Y'^} . Problems connected with 

 two confocal ellipses are also considered, and the initial motions of the 

 inner, due to any sudden motions impressed on the outer, are found. 



Coates '- has worked out the values of \p for a vortex outside and 



' ' Fluid motion in a rotating rectangle, formed by two concentric circular arcs 

 and two radii,' 3Iess. Math. ix. p. 35 (1879). 



- ' Solution by means of elliptic functions of some problems in the conduction of 

 electricity and heat in plane ligures,' Quart. Jour. xvii. p. 284 (1881). 



» Ibid. 



•* ' Plane vortex motion,' Quart. Jour. xv. p. 10 (1877). 



' 'On .some prolilems in the conduction of electricity,' Quart. Jour. xvii. -p. 65; 

 also Brit. Assoc. Brjj. (1879) p. 261. 



'* ' On the steady motion of incompressible fluids,' Cajuh. Phil. Trans, vii. p. 439, 

 and Reprint, vol. i. p. 10. 



'■ ' Sui principii fondaraentali dell'idrodinamica razionale,' Mem. di Bologna, iii. 



** ' On the motion of a mass of water about a moving cylinder,' Quart, jour. xiii. 

 p. 115. 



" ' Rome hydrodynamical solutions,' Quart. .lour. xiv. p. 40. 



'» ' Notes on Hydrodynamics,' ii. Me.ss. Math. ix. p. 117 (1880). 



" ' Fluid motion between confocal elliptic cylinders, &:c.,' Quart. Jour. xvi. p. 227 

 (1879). 



'2 'Vortex motion n and about elliptic cylinders,' Quart. Jour. xy. p. 356, and 

 xvi. p. 81 (1878.) 



