34 Mr. 0. Heaviside. On the Transformation of [Jan. 18, 



= 0, (6) 



8/18 C 3 8/(S 



where S//8 is as in (4), with / written for c. Similarly as regards the 

 second form of (3). When the operators are not homologous, the 

 complication of the form of the constituents of the inverse operators 

 makes the expansion less easy. 



As regards the second form of (3), it is obtained from the first 

 form by interchanging ft. and c. It represents the same surface. The 

 transformation from one form to the other, if done by ordinary 

 algebra, without the use of vectors and linear operators, is very 

 troublesome in the general case. But in the electromagnetic theory 

 the equivalence can be seen to be true and predicted beforehand. 

 For consider the circuital equations (2). If we eliminate H, we 



obtain 



curl ft' 1 curl E = cE, (7) 



whilst if we eliminate E, we obtain 



curl c- 1 curl H = ^H. (8) 



These are the characteristic equations of E and H respectively in 

 a dielectric with duplex eolotropy, and we see that they only differ 

 in the interchange of c and p. When, therefore, we apply one of 

 them, say that of E, to a plane wave to make the velocity equation, 

 in which process E is eliminated, we can see that a precisely similar 

 investigation applies to the H equation, provided // and c be inter- 

 changed. So, if the E equation leads to the first form in (3), the H 

 equation must lead to the second form. They therefore represent the 

 same surface. The same property applies to any equation obtained 

 from the circuital equations with the electrical variables eliminated, 

 the equation of the wave-surface, for example. If we have obtained 

 one special form, a second is got by interchanging the eolotropies. 



The index equation being what we are naturally led to from the 

 characteristic equation, it is merely a matter of mathematical work 

 to derive the corresponding wave-surface. For 8 is the reciprocal 

 of the perpendicular upon the tangent plane to the wave-surface, so 

 that 



= 1, (9) 



if r is the vector of the wave-surface ; and from the equation of 8 

 and its connexion with r, we may derive the equation of r itself. I 

 have shown (loc. cit., vol. 2, pp. 12 16) that the result is expressed 

 by simply inverting the operators in the index equation. Thus, the 

 equation of the wave-surface is 



