Pulsating Spheres in a Liquid. 115 



Let a doublet of moment — fi' be placed at Q, where fA> fjJ. 

 If TQL = 0, TQ=/, the flow through TT' due to this is 



— 27TJH 1 sin 2 cf> . /r 1 . 



The equation of a surface ot revolution across which there 

 is no flow from the doublets at P and Q is 



27r(//, sin 2 . jr—fj) sin 2 <]> . /r) = A, a constant. 



If A=0, this reduces to the line OL and the sphere 



fJbr~ Z = yLtV -3 . 



If C be the centre of this sphere and a the radius, 

 CP . CQ = a 2 and a= (a 2 -l)PQ/2a, 



where u 3 =/jl//j,'. 



Since the liquid is supposed to be " perfect/' and the sphere 

 is part of a surface of flow, we may make a material sphere 

 occupy its place. 



Hence the effect of a sphere X (fig. 1) in disturbing the 

 flow from the doublet \x at P, distant/ from its centre, is the 

 same as if the sphere were removed and a doublet —pa?// 3 

 placed at Q. This doublet is called the image of //, in the 

 sphere. 



§ 3. To determine the image of a source fi at P. 



This source is equivalent to a line of doublets of intensity fi 

 extending from P to infinity along PO (fig. 1) . Its image is 

 therefore a line of doublets from Q to C. 



If fjid be the moment of a doublet distant a 2 /x from the 

 centre, —fjidx 3 /a d is the moment of its image. But the 

 lengths of corresponding parts of Pco and CQ are as a 2 : x 2 . 



Therefore the intensity of the image is , supposing 



the axis along QC ; or — , regarding it as along CQ. 



a 



Now a magnet of length I and unit section, whose intensity 

 of magnetization at a distance x from one end is Gx, is equi- 

 valent (so far as concerns external action) to a quantity of 

 " magnetic matter " CZ at the positive end, and a distribution 

 of line-density — C along the axis. 



Therefore, from the analogy between sources and magnetic 

 poles, the line of doublets is equivalent to a source pa// at Q, 

 and a line sink of strength —fi/a per unit length along CQ. 



§ 4. To find the image of a system of doublets PQ (fig. 2) 

 of intensity fix, where x is the distance from Q. 



Let P', Q' be the points inverse to P and Q. 



By § 2 the image is a system of doublets lying between 

 F and Q'. 



12 



