220 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



The electromagnetic forces between conductors carrying these currents 

 are proportional* to the expressions (2) with the signs reversed. Hence in 

 the hydrodynamical problem the forces on the solids are opposite to those 

 which obtain in the electrical analogue. In the particular case where the 

 fixed solids reduce to infinitely thin cores, round which the fluid circulates, 

 the current-sheets in question are practically equivalent to a system of 

 electric currents flowing in the cores, regarded as wires, with strengths *, * , ... 

 respectively. For example, two thin circular rings, having a common axis, 

 will repel or attract one another according as the fluid circulates in the 

 same or in opposite directions through themf. This might have been 

 foreseen of course from the principle of Art. 24. 



Another interesting case is that of a number of open tubes, so narrow as 

 not sensibly to impede the motion of the fluid outside them. If flow be 

 established through the tubes, then as regards the external space the extremi 

 ties will act as sources and sinks. The energy due to any distribution of 

 positive or negative sources m lt m 2 , ... is given, so far as it depends on the 

 relative configuration of these, by the integral 



taken over a number of small closed surfaces surrounding m lt w 2 , ... respec 

 tively. If 1? &amp;lt; 2 , ... be the velocity-potentials due to the several sources, the 

 part of this expression which is due to the simultaneous presence of m 1} m 2 is 



which is by Green s Theorem equal to 



. dd&amp;gt; 9 Try ..... 



d&amp;gt;, / 2 dS (m). 



1 dn 



Since the surface-integral of d&amp;lt;f&amp;gt; 2 /dn is zero over each of the closed surfaces 

 except that surrounding w 2 , we may ultimately confine the integration to the 

 latter, and so obtain 



Since the value of (f) 1 at m 2 is mjr 12 , where r 12 denotes the distance between 

 wij and ?n 2 , we obtain, for the part of the kinetic energy which varies with the 

 relative positions of the sources, the expression 



* Maxwell, Electricity and Magnetism, Art. 573. 



t The theorem of this paragraph was given by Kirchhofi, 1. c. ante p. 59. See 

 also Sir W. Thomson, &quot;On the Forces experienced by Solids immersed m a Moving 

 Liquid,&quot; Proc. R. S. Edin., 1870; Reprint, Art. xli. ; and Boltzmann, &quot; Ueber die 

 Druckkrafte welche auf Hinge wirksam sind die in bewegte Fliissigkeit tauchen,&quot; 

 Crelle, t. Ixxiii. (1871). 



