226 Mr. Nicholson on Electrical Vibrations 



dissipation of energy, and the surface conditions may all be 

 included in the .statements : — 



(1) The resultant magnetic induction at the surface is 

 tangential. 



(2) The resultant electric force is normal. 



In an oscillation, these vectors are perpendicularly situated 

 in the plane of the wave-front ; hence the latter plane is 

 tangential or normal to the surface. 



The problem is confined to the case of an elliptic cross- 

 section. If the section were parabolic or hyperbolic, the 

 energy would be radiated into space, and the oscillation could 

 not be permanent. If the axis of z be that of the cylinders, 

 and [xy] the plane of a section perpendicular to the axis, a 

 point of space is defined by (X/u,z), where (X, fi) are the roots of 



^ + -1^=1. • 



X b 2 + X 



A=const., /^ = const., being systems of elliptic and hyperbolic 

 cylinders respectively. 



The space elements in the directions X, p- are 



j dX -j dfj. 



2pi_ 2^ 2 



where _ /\(b 2 + X) 



m (say), 



p 2 = X/ 



P(aO 



X—f-i 



In the usual electrical notation, the current is denoted by 

 (u, v, w), the electric force by (X, Y, Z), and the magnetic 

 force and magnetic induction by (a, b, c) . 



The two latter vectors are identical, if the medium between 

 the two surfaces be non-magnetic. 



If V is the velocity of light, and the current is entirely 

 sethereal, 



4*-V»(u,r, w) = (X,T, Z) (1) 



By the circuital relation of Ampere, 



4-rru _ B , % B / o 



~dX \j) 2 / "dfiXpi/ _ 



2ttw_ !$_ ( b\ _$_( 



P1P2 



