48 EEPOET — 1882. 



method, as a rule, clocs not afford useful results, as tbo curves are in 

 general too complicated. The mathematical interest attaches itself to 

 solutions for the case of given boundaries or given conditions, and 

 reduces itself often to a suitable transfoi'mation by conjugate functions, 

 whereby the given boundary maybe transformed to one consisting of lines 

 or circles, the solution of which is known. This has been applied in 

 some of the preceding examples. Of direct solutions other than those 

 already mentioned two require notice here. One by Ferrers,' who has 

 determined the (]i and -^ functions for the spaces (1) inside an ellipse and 

 between the two branches of a confocal hyperbola, and (2) between an 

 ellipse and one branch of a confocal hyperbola, when the boundary 

 rotates; also for two confocal parabolas, the limiting case of (2). The 

 functions ai-e given in infinite series. The other is by Greenhill.^ He 

 has investigated expressions for the f and i^, due to a source, a doublet, 

 or a vortex, in the space bounded by Cartesians, in terms of the con- 

 jugate functions given by a; + ii/ = sn"^(| + iij), in which £,»/ give the 

 confocal Cartesians, whose vectorial equations are r' — rdn4' = cuE, and 

 r' + rdniij = cn»/, the foci being at the points x = o, 1, 1//.-^ and y =■ o. 



Non-plane Two-dimensional Motion. — The hydrodynamical interest of 

 plane two-dimensional motion consists in its physical application to the 

 motion of cylinders in an infinite fluid, or of cylinders of finite length in 

 the space between two planes perpendicular to and touching the ends of 

 the cylinder. When the space considered is not plane, the motion may 

 be represented physically by the steady motion of electricity, the surface 

 being supposed a conductor. The surface of a s^iherc is one which has 

 received some attention. The case of the motion for a source and sink 

 at opposite extremities of a diameter was discussed by Robertson Smith ^ 

 in his paper referred to above. Beltrami* has given the general 

 solution of the equation of continuity for the surface of a sphere which is 

 analogous to that for plane space in terms of polar co-ordinates. He 

 shows that, d and (p being the co-latitude and longitude of a point, the 

 general solution for the potential is in the form 



(a log tang + B j (af -f &) 4- 2 i Antan^^ 2 + B„cot" ^ jcos(?i^ 4- a). 



This he applies to the case of the motion of a spherical cap on the sphere, 

 and finds that, a being the spherical radius of the cap and (6q . ^tt) the 

 co-ordinates of the instantaneous centre of rotation, the potential is pro- 

 portional to sin fio(sin ^aycot ^0 cos <j>, the centre of the cap being at the 

 pole. The lines of flow are given by cot ^9 sin (j) = const., and of flow 

 relative to the cap by (cos a — cos 0) cot ^0 sin f + cot 0^ cos = const. 

 These are the intersections with the sphere of hyperbolic cylinders 

 whose asymptotic planes are, one parallel to the boundary of the cap (a 

 small circle of the sphere), and the other to the great circle of the sphere 

 which gives the instantaneous direction of motion of the centre of the 

 cap. He also discusses the time a particle takes to describe its path, and 

 particular forms of the paths. The f and \p functions for sources and 

 sinks on a sphere, or certain portions of spheres, bounded by circles, have 



' ' On the motion of water contained in certain cylindrical vessels,' Quart. Jour. 

 xvii. p. 227 (1880). 



- ' On functional images in Cartesians,' Quart. Jour, xviii. p. 231, 846 (1882). 



3 ' On the flow of electricity, &c.' Proc. Boy. Soo. Edin. vii. p. 79 (1870). 



■* 'Intorno ad un caso di moto a due co-ordinate.' Jiendiconti d. reale, 1st. 

 Lovib. (II.) xi. p. 199 (1878). 



