sinh at 



Equations in a Homogeneous Isotropic Medium. 43 



and so on, in a uniform manner, thus proving that the suc- 

 cessive terms of (48) are the successive terms of the expansion 

 of (45) (right number) in powers of o-^ j and therefore 

 proving (45). 



The following formulse occur when the front of the wave is 

 in question, where caution is needed in evaluations : — 



cosho-i— i=-l coshgicZm, . . (49) 



2 r* &va.m,vt Bvakat -, ,^r.\ 



= - 1 ^ dm,. . . (50) 



7rJ„ m q 



15. Interpretation of Results. — Having now given a con- 

 densation of the mathematical work, we may consider, in 

 conclusion, the meaning and application of the formulae. In 

 doing so, we shall be greatly assisted by the elementary theory 

 of a teleo-raphic circuit. It is not merely a mathematically 

 analoD-ous theory, but is, in all respects save one, essentially 

 the same theory, physically, and the one exception is of a 

 remarkable character. Let the circuit consist of a pair of 

 equal parallel wires, or of a wire with a coaxial tube for the 

 return and let the medium between the wires be slightly 

 conducting. Then, if the wires had no resistance, the problem 

 of the transmission of waves would be the above problem of 

 plane waves in a real dielectric, that is, with constants /a, c, 

 and k, but without the magnetic conductivity; i. e. ^=0 in 

 the above. 



The fact that the hues of magnetic and electric force are no 

 longer straight is an unessential point. But it is, for conve- 

 nience, best to take as variables, not the forces, but their 

 line-integrals. Thus, if Y be the line-integral of E across the 

 dielectric between the wires, V takes the place of E. Thus 

 /<:E the density of the conduction-current, is replaced by 

 KV where K is the conductance of the dielectric per unit 

 length of circuit, and cE/47rj the displacement, becomes SV, 

 where S is the permittance per unit length of circuit. The 

 density of electric current cpE/47r is then replaced by SpV. 

 Also tS V is the charge per unit length of circuit. 



Next, take the line-integral of H/47r round either conductor 

 for magnetic variable. It is C, usually called the current in 

 the wires. Then /iH, the induction, becomes LC ; where LG 

 is the momentum per unit length of circuit, L being the 

 inductance, such that LSv^=/xcv^==l. 



A more convenient transformation (to minimize the trouble 

 with 47r's) is 



^^^X' ^ to 1^5 ^toS, 47r^toK. 

 sx to \J J 



