considered in connexion with the Dynamical Theory. 7 



Or we may take an illustration, which is in many respects to be 

 preferred, from the disturbance of a perfect fluid, by the motion 

 of solid bodies iu its interior. Thus if in an infinite fluid two 

 spheres move parallel to each other and perpendicularly to the 

 line joining them, and with such small velocities that their rela- 

 tive position does not sensibly change, the kinetic energy may 

 as usual be expressed by 



x,y denoting the velocities of the two spheres, and L,M,N being 

 approximately constants*. When the spheres move in the same 

 direction, the reaction of the fluid tends to press them together ; 

 but if the motions are opposed, the force changes to a repulsion. 

 We see here the analogues of the phenomena of attraction and 

 repulsion discovered by Ampere. If when all is at rest a given 

 velocity is impulsively impressed on one sphere, the other im- 

 mediately starts backwards, and, as Thomson f has shown, with 

 uch velocity that the energy of the whole motion is the least 

 possible under the given condition. 



This theorem is general, and leads directly to the solution of 

 a large class of electrical problems connected with induction ; 

 for whenever a current is suddenly generated in one of the 

 circuits of a system, the initial currents in all the others are 

 to be determined so as to make the energy of the field a mini- 

 mum. These initial currents are formed unmodified by resist- 

 ance whenever the electromotive impulses to which they owe 

 their existence last only for a time which may be regarded as va- 

 nishingly small compared with the time-constants of the circuits. 

 The sudden fall of a current when a circuit is opened generates 

 the same currents, except as to sign, in neighbouring circuits as 

 those due to a rise of the first current, and the condition as to 

 sufficient suddenness is more generally fulfilled ; at the same time 

 it is more convenient in explaining the theory to take the case of 

 the establishment of the primary current. 



Suppose, then, that in the wire A 2 of our coil a current x is 

 suddenly generated, while the ends of A 2 are joined by a short 

 wire. The condition of minimum energy is obviously fulfilled 

 if there arise in A 2 a current represented by— x ; for then the 

 energy of the field is approximately zero. But if the self-induc- 

 tion of the wire joining the ends of A 2 be sensible, the annihi- 

 lation of the energy can no longer be perfect. Thus, let the 

 circuit of A 2 be completed by B 1 B 2 , then the general expression 

 for the energy of two currents becomes in this case 

 1L# 2 + Lxy + 1L/ x (5 or 1, 



* Thomson and Tait's ' Natural Pkilosopkv/ pp. 262, 264. 

 t Thomson and Tait, p. 225. 



