42 REPORT— 1882. 



a line can be deduced from the case of the ellipse referred to below. It 

 is of importance from an electrical point of view, but is only of purely 

 mathematical interest as far as hydrodynamics is concerned. 



Passing on now to spaces bounded by straight lines, we have to notice 

 the cases of two parallel lines, triangles, and rectangles. Sources and 

 sinks between parallel planes have been discussed by the writer, but they 

 may be regarded as limiting cases of the rectangle, which is referred to 

 below. When the triangle is equilateral, the motion for a rotation was 

 discovered by F. D. Thomson,' and Stokes^ has shown that for this the 

 effective moment of inertia of the equivalent solid is two-fifths of that of 

 the solidified fluid. The potential and stream functions for a vortex inside 

 Buch a triangle have been given by Greenhill,^ and the path described by 

 the vortex, also the same functions when there is a source at one corner 

 and a sink at the other. For the right-angled isosceles triangle the case 

 for a source and sink at the base angles have been given by the writer,'' 

 and are, I believe, the only ones solved for this triangle. If the vertex of 

 a triangle move off to infinity jierpendicular to the base, we get the space 

 bounded on three sides by two infinite parallel lines, and a third line per- 

 pendicular to them. The potential and stream function for a source, or a 

 vortex (electric point) in such a space are given in the same paper, as well 

 as figures illustrating them in particular cases. This may be regarded as 

 a limiting case of the rectangle, to which we now proceed. 



This form has received much attention from the time when Stokes^ 

 first discussed it in ] 843. He determined the velocity potential for the 

 internal motion when the boundary rotates, in the form of an infinite series, 

 and in the same form evaluated the moment of inertia of the equivalent 

 solid. His method is based on determining the coefficients of an infinite 

 series to represent the motion of rotation of two opposite sides when the 

 other pair are at rest, and then combining them for the whole motion. 

 The same problem is considered in * Thomson and Tait's Natural Philo- 

 sophy.' ^ Ferrers^ (1878) attacks the question differently. He first 

 shows that if the density of matter at every part of a plane be given by 

 p cos mx cos vy, its potential is 4<7rp cos vix cos ny /(m^ + n^). Now the 

 analytical conditions for the stream function of the motion of fluid in a 

 rotating rectangle are the same as for the potential for a distribution of 

 matter of density + 1, and — 1, in alternate equal rectangles. This 

 density is expressed as the product of two Fourier's series, and the before- 

 mentioned theorem applied. Greenhill* (1878) gives expressions for the 

 velocities at any point in a very compact form as definite integrals of 

 certain elliptic functions of the position of the point. He arrives at this 



' ' On certain cases of fluid motion,' Ox. Cam. and Dub. Mess. Math. iii. p. 238. 

 ^ Reprint of papers, p. 65. 



' ' Solutions by means of elliptic functions of some problems in the conduction of 

 electricity and of heat in plane figures,' Qua7't. Jour. xvii. p. 28-t. 



* ' On velocity and electric potentials between parallel planes,' Qiuwt. Jour. xv. 

 p. 313. 



* ' On some cases of fluid motion,' Trans. Camh. Phil. Soc. viii. p. 105 (1843). 

 Supplement to a memoir ' On some cases of fluid motion' (1846), Ibid. p. 409. ' On 

 the critical values of the sums of periodic series' (1849), Ibid. p. 533, sec. iv. In 

 the reprint of papers these are vol. i. pp. 60, 188, and 288. 



' Vol. i. 1st ed. p. 541. 



' ' Solution of certain questions in potentials and motion of liquids,' Quart. Jour, 

 XV. p. 83. 



" ' Notes on hydrodynamics : on the motion of water in a rotating rectangular 

 prism,' Quart. Jour. xv. 144. 



