Forced Vibrations of Electromagnetic Systems. 131 



An auxiliary equation to exclude unipolar magnets, viz. 



div. B = 0, (5) 



expressing that B has no divergence. The most important 

 feature of this scheme is the equation (8), as a fundamental 

 equation, the natural companion to (2). 



The derived energy relations are not necessary, but are 

 infinitely too useful to be ignored. The electric energy U, 

 the magnetic energy T, and the dissipativity Q, all per unit 

 volume, are given by 



U = |ED, T = ihB/4tt, Q = EC. . . . (6) 



The transfer of energy W per unit area is expressed by a 

 vector product, 



W = V(E-e)(H-h)/47r, .... (7) 



and the equation of activity per unit volume is 



eF + hG=Q + U + T + div.W, .... (8) 

 from which W disappears by integration over all space. 



The equations of propagation are obtained by eliminating 

 either E or H between (2) and (3), and of course take different 

 forms according to the geometrical coordinates selected. 



In a recent paper I gave some examples * illustrating the 

 extreme importance of the lines of vorticity of the impressed 

 forces, as the sources of electromagnetic disturbances. Those 

 examples were mostly selected from the extended develop- 

 ments which follow. Although, being special investigations, 

 involving special coordinates, vector methods will not be used, 

 it will still be convenient occasionally to use the black letters 

 when referring to the actual forces or fluxes, and to refer to 

 the above equations. The German or Gothic letters employed 

 by Maxwell I could never tolerate, from inability to distin- 

 guish one from another in certain cases without looking very 

 hard. As regards the notation EC for the scalar product of 

 E and C (instead of the quaternion —SEC) it is the obvious 

 practical extension of EC, the product of the tensors, what 

 EC reduces to when E and C are parallel, f 



* Phil. Mag. Dec. 1887, " On Resistance and Conductance Operators," 

 § 8, p. 487. 



t In the early part of my paper "On the Electromagnetic Wave- 

 Surface," Phil. Mag. June 1885, I have given a short introduction to the 

 Algebra of vectors (not quaternions) in a practical manner, i. e. without 

 metaphysics. The result is a thoroughly practical working system. The 

 matter is not an insignificant one, because the extensive use of vectors iu 

 mathematical physics is bound to come (the sooner the better), arid my 

 method furnishes a way of bringing them in without any study of 

 Quaternions (which are scarcely wanted in Electromagnetism, though 

 K 2 



