A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 537 



In the case of electric currents, the resistance to sudden increase or dimi- 

 nution of strength produces effects exactly like those of momentum, but the 

 amount of this momentum depends on the shape of the conductor and the 

 relative position of its different parts. 



Mutual Action of two Currents. 



(23) If there are two electric currents in the field, the magnetic force at 

 any point is that compounded of the forces due to each current separately, 

 and since the two currents are in connexion with every point of the field, 

 they will be in connexion with each other, so that any increase or diminution 



of the one will produce a force acting with or contrary to the other. 



/ 



Dynamical Illustration of Reduced Momentum. 



(24) As a dynamical illustration, let us suppose a body C so connected 

 with two independent driving-points A and B that its velocity is p times that 

 of A together with q times that of B. Let u be the velocity of A, v that 

 of B, and w that of C, and let Sx, Sy, Sz be their simultaneous displacements, 

 then by the general equation of dynamics*, 



where X and Y are the forces acting at A and B. 



dw du dv 



But dt = Pdi + ^Tt' 



and Sz = p8x + q&y. 



Substituting, and remembering that Sx and Sy are independent, 





, ............................ (I)- 



We may call Cp'u+Cpqv the momentum of C referred to A, and Cpqu + Cq*v 

 its momentum referred to B ; then we may say that the effect of the force 

 X is to increase the momentum of C referred to A, and that of Y to increase 

 its momentum referred to B, 



* Lagrange, Mec. Anal. II. 2, 5. 



VOL. i. 68 



