200 Lord Rayleigh on the Propagation of Electric 



return. For practical applications it is essential to treat the 

 conductivity of the wire as finite ; but for some scientific 

 purposes the conductivity may be supposed perfect without 

 much loss of interest. Under this condition the problem is 

 so much simplified that important extensions may be made 

 in other directions. For example, the complete solution may 

 be obtained for the case of parallel wires, even although the 

 distance between them be not great in comparison with their 

 diameters. 



We may start from the general equations of Maxwell 

 involving the electromotive intensity (P, Q, R) and the mag- 

 netic induction (a, b, c), introducing the supposition that all 

 the functions are proportional to e i(j>t+Tnz \ and further that 

 m=p/V, just as in the case of uninterrupted plane waves 

 propagated parallel to z. Accordingly d 2 /dt 2 = Y 2 d 2 /dz 2 , and 

 any equation such as 



72 P J2T> 1 .72 \J 



(1) 



(2) 



reduces to 



d 2 V d 2 ? d 2 P 1 d 2 ¥ 



da? + dy 2 + dz 2 ~ V 2 dt 2 



d 2 ? ,d 2 P _ Q 



. . . 





dx 2 dy 2 





+ J^)(P, Q, R,a,6,c)=0. ... (3) 



They may be summarized in the form 



/d^ . d 2 

 \dx 2 



The case to be here treated is characterized by the con- 

 ditions R=0, c = ; but it would suffice to assume one of 

 them, say the latter. Since in general throughout the 

 dielectric 



dc/dt = dF/dy-dQ/dx, . . . ~ . (4) 



it follows that P and Q are derivatives of a function (<£), also 

 proportional to e i< - pt+ "' z \ which as a function of x and y may 

 be regarded as a potential since it satisfies the form (2). 

 Thus dP/dx + dQ,/dy = 0, from which it follows that dRJdz 

 and R vanish. It will be convenient to express all the func- 

 tions by means of (j>. We have at once 



Y = d$ldx, Q = d<j>/dy, R=0. . . . (5) 



Again, by the general equation analogous to (4), since R = 0, 

 ipa = imQ, ; so that 



a = Y- 1 d(f>/dy, b^-Y-'dcp/dx, c = 0. . (6) 



Thus the same function <£ serves as a potential for P, Q and 

 as a stream-function for a, b. 



