141] MUTUAL ACTION OF SOURCES. 221 



The quantities m x , w 2 , ... are in the present problem equal to l/4?r times 

 the fluxes , xo j across the sections of the respective tubes, so that (v) 

 corresponds to the form T of the kinetic energy. The force apparently ex 

 erted by wij on w 2 , tending to increase r 12 , is therefore, by (8), 



d 



Hence two sources of like sign attract, and two of unlike sign repel, with 

 forces varying inversely as the square of the distance*. This result, again, 

 is easily seen to be in accordance with general principles. It also follows, 

 independently, from the electric analogy, the tubes corresponding to Ampere s 

 solenoids. 



We here take leave of this somewhat difficult part of our 

 subject. To avoid the suspicion of vagueness which sometimes 

 attaches to the use of generalized coordinates/ an attempt has 

 been made in this Chapter to make the treatment as definite as 

 possible, even at some sacrifice of generality in the results. There 

 can be no doubt, for example, that with proper interpretations the 

 equations of Art. 137 will apply to the case of flexible bodies 

 surrounded by an irrotationally moving fluid, and even to cases of 

 isolated vortices (see Chap, vn.), but the justification of such 

 applications belongs rather to general Dynamics-)-. 



* Sir W. Thomson, Reprint, Art. xli. 



t For further investigations bearing on the subject of this Chapter see J. Purser, 

 &quot;On the Applicability of Lagrange s Equations in certain Cases of Fluid Motion,&quot; 

 Phil. Mag., Nov. 1878 ; Larmor, I.e. ante p. 210 ; Basset, Hydrodynamics, Cam 

 bridge, 1888, c. viii. 



