378 Mr. 0. Heaviside on Electromagnetic Waves, and the 



number of concentric shells, each being homogeneous, and 

 having special values of c, k, jju, and g. For the spherical 

 functions would not be suitable otherwise, except during 

 the passage of the primary waves to the boundaries, or until 

 they reached places where the departure from the assumed 

 constitution commenced. Assuming the constitution in homo- 

 geneous spherical layers, there is no difficulty in building up 

 the forms of y and y x in a very simple and systematic manner, 

 wholly free from obscurities and redundancies. In any 

 layer the form of E/H is as in (284), containing one y. Now 

 at the boundary of two layers E is continuous, and also H 

 (provided the physical constants are not infinite), so E/H is 

 continuous. Equating, therefore, the expressions for E/H in 

 two contiguous media expresses the y of one in terms of the 

 y of the other. Carrying out this process from the origin 

 up to the medium between a and a, expresses y in terms of 

 the y of the medium containing the origin ; this is zero, so 

 that y is found as an explicit function of the values of u, iv, 

 u f , w' at all the boundaries between the origin and a . In a 

 similar manner, since the y of the outermost region, extend- 

 ing to infinity, is 1, we express y 1? belonging to the region 

 between a and a u in terms of the values of w, &c. at all the 

 boundaries between a and co . Each of these four functions 

 will occur twice for each boundary, having different values of 

 the physical constants with the same value of r. I mention 

 this method of equation of E/H operators because it is a far 

 simpler process than what we are led to if we use the vector 

 and scalar potentials; for then the force of the flux has three 

 component vectors — the impressed force, the slope of the scalar 

 potential, and the time-rate of decrease of the vector potential. 

 The work is then so complex that a most accomplished mathe- 

 matician may easily go wrong over the boundary conditions. 

 These remarks are not confined in application to spherical 

 waves. 



If an infinite value be given to a physical constant, special 

 forms of boundary condition arise, usually greatly simplified ; 

 e. g. infinite conductivity in one of the layers prevents electro- 

 magnetic disturbances from penetrating into it from without ; 

 so that they are reflected without loss of energy. 



Knowing y 1 and y in (288), we virtually possess the sinu- 

 soidal solution for forced vibrations, though the initial effects, 

 which may or may not subside or be dissipated, will require 

 further investigation for their determination; also the solu- 

 tion in the form of an infinite series showing the effect 

 of suddenly starting / constant; also the solution arising 

 from any initial distribution of E and H of the kind appro- 



