36 Mr. 0. Heaviside on Maxwell's Electromagnetic 



impressed forces. The remarks in this paragraph are not 

 limited to homogeneity and isotropy. 



9. Solutions for Plane Waves. — If we take z normal to the 

 plane of the waves, we may suppose that both E and H have 

 X and y components. This is, however, a wholly unnecessary 

 mathematical complication, and it is sufficient to suppose that 

 E is everywhere parallel to the a;-axis and H to the y-axis. 

 The specification of an initial state is therefore Eq, Hq, the 

 tensors of E and H, given as functions of z; and the equations 

 of motion (1), (2) become 



Now the operator q^ in (5) becomes 



/=v2y' + o-2; (16) 



where by V we may now understand d/dz simply. Therefore, 

 by (7), the solutions of (15) are 



E = e-pf fcosh qt- ^ sinh qt)^,- ^^^^ hJ . ) 



L^ ^ ^ .1 ' -^ 1(17) 



H = e-P^[(cosh qt+ ^ sinh qt)B,- ^i^^^Eo]. ) 



When the initial states are such as ae^^, or a cos bz, the reali- 

 zation is immediate, requiring only a special meaning to be 

 given to q in (17). But with more useful functions as ae~*^^, 

 &c., &c., there is much work to be performed in effecting the 

 differentiations, whilst the method fails altogether if the initial 

 distribution is discontinuous. 



But we may notice usefully that when Eq and Hg are 

 constants the solutions are 



E = e-2pi'Eo, H = e-2p2^Ho, . . . (18) 



which are quite independent of one another. Further, since 

 disturbances travel at speed v, (18) represents the solutions in 

 any region in which Eq and Hq are constant, from ^ = up to 

 the later time when a disturbance arrives from the nearest 

 plane at which Eq or Hq varies. 



10. Fourier Integrals. — Now transform (17) to Fourier 

 integrals. We have Fourier's theorem, 



f{z)— — \ I f {a) COS m{z— a) dm da, . . (19) 



