DECENT PEOGKESS IN HYBRODYNAMICS. 51 



outside a sphere has been given by myself ' (1879), and may be thus 

 stated. If a source, of strength ;h, is placed at a distance r, from the 

 centre of a sphere of radius a, the 'image' consists of (1) a source at 

 the optical image of m, and of magnitude majr ; and (2) a constant 

 line-sink extending from this isolated image to the centre, and of line- 

 density, mja. The image of a doublet (/u) with its axis in any direction 

 is then easily deduced by making a source and sink approach indefinitely 

 near to one another. When the axis points towards the centre, it is 

 clear that the line-distribution disappears, and we get the result found by 

 Stokes ; when the axis is perpendicular to the line joining it to the 

 centre, the image consists of an isolated doublet ^ /xa^'/r^ at the inverse 

 point, and a line doublet thence to the centre, whose line-density at a 

 distance p is — fipjar. It is curious ^ that if there is a source at a point 

 P and a constant line-sink betAveen P and Q, where Q is a point on the 

 line from P to the centre, then, provided the whole amount of the line- 

 sink is equal to the amount of the source, the * image ' of this arrange- 

 ment is an arrangement of the same form — viz., a source at P' the inverse 

 of P, and a constant line-sink between P' and Q', the inverse of Q, the 

 whole amount of the line-sink being equal to that of the source at P'. 

 This is of importance in the treatment of the motion of two sjDheres, 

 when one at least changes its volume. 



The image of another kind of singular point, that of the element of a 

 vortex filament, has been determined by Lewis. ^ In this case the image 

 of a small element of a vortex line is the optical image of the element, 

 and their strengths are iuvei'sely proportional to the square roots of their 

 distances from the centre. Hence, any complete vortex filament has a 

 complete image, provided it lies on a concentric sphere. By means of 

 this theorem Lewis has investigated the motion of a circular vortex fila- 

 ment inside a sphere when it moves symmetrically with respect to a 

 diameter. When it occupies a position of rest, its radius (r) is given by 

 (0,2 _ }-2^ jQg f3j^j.3^Q,2 _ r'^yi^a^h^} —■ 4a^ ; where a is the radius of the 

 sphere, and b is the radius of the sphere whose volume is equal to that of 

 the filament. 



The motion standing next in simplicity is that of the initial motion of 

 the fluid contained between two concentric spheres when the inner begins 

 to move. This forms one of the examples considered by Stokes * in his 

 paper ' On some cases of fluid motion' (1843). He first finds the velocity 

 potential for any motion of the bounding surfaces, and shows that if the 

 inner sphere performs a small oscillation within the outer as a fixed boun- 

 dary, the motion is the same as if the inertia be supposed increased by a 

 mass equal to ^(P + 1a^)l{h^ — a^) times the mass of the fluid displaced, 

 where a, h denote the radii of the inner and outer spheres respectively. 

 He then passes on to the case where a sphere moves in fluid bounded by 

 an infinite plane. This is important as the first application in hydrody- 

 namics of the principle of successive ' reflection ' of motion. Taking first 

 the motion of the sphere perpendicular to the plane, he finds the normal 

 motion at points of the plane due to the motion of the sphere on the 



' ' On the motion of two spheres in a fluid,' Tram. Boy. Soc. part ii. (1880), 

 p. 455. 



^ ' On the problem of two pulsating spheres in a fluid,' Proc. Camb. Phil. Soc. 

 iii. p. 276. 



' ' On the images of vortices in a spherical vessel,' Quart. Jour, xvi, p. 338 (1879). 



* Trans. Camb. Phil. Soc. viii. 105. 



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