EECENT rEOGtRESS IN HTDnODYNAMICS. 49 



been investigated by Hill,' who arrives at the necessary expressions by 

 transforming the variables so as to make the equation of continuity of 

 the same form as in plane motion, and taking similar functions of those 

 variables. Amongst many interesting results may be mentioned those 

 for equal source and sink. Hero the stream lines ai'e the small circles 

 through the source and sink (intersections of sphere with planes through 

 the chord joining the points), and the potential lines are the system of 

 small circles orthogonal to the foregoing (the intersections with planes 

 passing through the line of intersection of the tangents to the sphere 

 at the source and sink). Allen '^ has made a valuable remark, that the 

 transformation which Hill has used is geometrically equivalent to trans- 

 forming the equipotential and stream lines for any motion on a plane by 

 a stereographical projection into a corresponding motion on the sphere. 

 This might be regarded as making the theory for the spherical surface as 

 complete as for the plane, were it not that the projections do not always 

 correspond in simplicity to the original curves. For instance, he shows 

 that confocal conies project into quartic curves, and that confocal sphero- 

 conics are the projections of quartic curves. The case of motion on 

 the surface of a cylinder is also touched upon, as it has also been by 

 Boltzmann.^ 



b. Motion in three dimensions. 



Planes. — The image of a source in presence of an infinite plane has 

 long been known, and is obvious. Stokes was, I believe, the first to 

 employ it. The velocity-potential for a source between two parallel 

 planes — which is the sum of the same functions for the infinite train of 

 images — has been given by myself,* whilst Greenhill ^ has solved the 

 corresponding problem for the case of a rectangular box. When the 

 origin is at one corner and a single source at the point Xi-y^.z^ the poten- 

 tial is 



TT 



H ttH 



+ similar terms in i/j, z-^. 



In any actual case this has to be combined with an equal sink at some 

 other point ; it gives the power of solving the general problem of any 

 motion of the sides, and has been used by Kirchhofi" to determine the 

 electrical resistance of a conducting parallelepipedon. 



The Sphere. — The velocity-potential for the sphere was first found by 

 Poisson ^ in 1831, in discussing the effect of the air on the motion of a 

 ball-pendulum in it. If the elasticity of the air be neglected we get the 



' ' The steady motion of electricity in spherical current sheets,' Quart. Jovr, 

 xvi. p. 306 (1879). 



* ' On some problems in the conduction of electricity,' Qtiart. Jour. xvii. p. 65 

 (1880). 



' ' Bewegung cler Electricitat auf einer cylindrischen Flache,' Wien. Sitzher. Hi. 

 (2) p. 220. 



■* ' On velocity and electric potentials between parallel planes,' Quart. Jov/r. 

 XV. p. 293 (1878). 



* ' On Green's functions for a rectangular parallelepiped,' Proc. Cnmh. Plnl. Soc. 

 iii. p. 289. 



* ' Memoire sur los mouvemcnts simultanfis d'uu pendule et tie I'air environnant. 

 Mem. (le VAcatl. d. Sc. Paris, xl. p. 621 (1832). 

 1882. E 



