Equations in a Homogeneous Isotropic Medium, 41 



The results bring the Fourier integrals (21) to 



H = e-p' [i(Ho + cvF^Q)z-vt + KHo-cvEo).+.i 



)■ ■ (40) 



where 



P 



=:dldt, S7=d/dz, 2j=-{{z-a)^-vH^\K 



Another interesting form is got by the changes of variables 



(41) 



These lead to 



U =^epXE—fiv'R), u=z—vt,l 



t.i 



W = ieP^(E + yLit'H), io=z + vt. 



U;,,i=U^,o +1 (Uo^-^WojJo|^(?^-a)^(w-a)*|^a, 



WM=W,,o-r(Wo|^ +|5Uo)jo|^ iu-a)i{w-a)ijda. 

 The connexions and partial characteristic of U or W are 



and this characteristic has a solution 



U = (S)^J»[^(^^-»^'^)*],- • • (44) 



where m is any + integer, and in which the sign of the ex- 

 ponent may be reversed. We have utilized the case m = 

 only. 



14. Evaluation of Fourier Integrals. — The effectuation of 

 the integration (direct) of the original Fourier integrals will 

 be found to ultimately depend upon 



y\osm^SI^im=lj„[2(.^-„^i=)i], . (45) 



provided vt > z, where, as before, 



5^ = cr^ — nv^v^. 



