138 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



i.e. the surface is a sphere in relation to which P 1 and P 2 are 

 inverse points. If be the centre of this sphere, and a its radius, 

 we readily find 



.................. (3). 



This sphere may evidently be taken as a fixed boundary to the 

 fluid on either side, and we thus obtain the motion due to a 

 double-source (or say to an infinitely small sphere moving along 

 Ox) in presence of a fixed spherical boundary. The disturbance 

 of the stream-lines by the fixed sphere is that due to a double- 

 source of the opposite sign placed at the inverse point, the ratio 

 of the strengths being given by (3)*. This fictitious double- 

 source may be called the image of the original one. 



96. Rankine employed -f- a method similar to that of Art. 71 

 to discover forms of solids of revolution which will by motion 

 parallel to their axes generate in a surrounding liquid any given 

 type of irrotational motion symmetrical about an axis. 



The velocity of the solid being u, and 8s denoting an element 

 of the meridian, the normal velocity at any point of the surface is 

 udvr/ds, and that of the fluid in contact is given by d^/^ds. 

 Equating these and integrating along the meridian, we have 



v/r = - lutzr 2 + const ...................... (1). 



If in this we substitute any value of ^ satisfying Art. 93 (3), we 

 obtain the equation of the meridian curves of a series of solids, 

 each of which would by its motion parallel to x give rise to 

 the given system of stream-lines. 



In this way we may readily verify the solution already obtained 

 for the sphere ; thus, assuming 



+ = Au*/r* ........................... (2), 



we find that (1) is satisfied for r = a, provided 



A = -ua* .......................... (3), 



which agrees with Art. 95 (1). 



* This result was given by Stokes, &quot; On the Resistance of a Fluid to Two Oscil 

 lating Spheres,&quot; Brit. Ass. Report, 1847 ; Math, and Plnjs. Papers, t. i., p. 230. 



t &quot;On the Mathematical Theory of Stream-Lines, especially those with Four 

 Foci and upwards,&quot; Phil. Trans. 1871, p. 267. 



