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



or, in full, referring to the forms of u and w, equations (110) 

 to (113), 



H«»{rM(i.i)(i + I) 

 z/j, vr (. \ qa/ \ qrJ 



+e -^.,( 1+ i)( 1+ I)| /l . . (189) 



Effect the integrations indicated by the inverse powers of q 

 or p/v ; thus 



if /i be zero before and constant after t = 0. As for the 

 exponentials, use Taylor's theorem, as only differentiations 

 are involved. We get, after the process (140) has been 

 applied to (139), and then Taylor's theorem carried out, 



2/j,oVr \\ a r tar J \ a r 2ar J J 



where , , . 



rt l = vt — r + a } 



■ rt — r 



II 



It is particularly to be noticed that the t x part of (141) only 

 comes into operation when t x reaches zero, and similarly as 

 regards the f 2 part. Thus, the first part expresses the primary 

 wave out from the surface; the second, arriving at any point 

 2a/r later than the first, is the reflected wave from the centre, 

 arising from the primary wave inward from the surface. 

 The primary wave outward may be written 



-G©S( 1 ^ • • • (-> 



where vt>(r—a), and the second wave by its exact negative, 

 with vt >(r + a). Now, by comparing (io2) with (134), we 

 see that the internal solution is got from the external by 

 exchanging a and r in the -j j-'s in (139) and (141), including 

 also in / x and t 2 . The result is that (142) represents the 

 internal H in the primary inward wave, vt having to be 

 >(a — r); whilst its negative represents the reflected wave, 

 provided vt > (a + r). 



The whole may be summed up thus. First, vt is <a. 

 Then (142) represents H everywhere between r=a + rt and 

 r = a — vt. But when vt is >a, H is given by the same 

 formula between the limits r—vt—a and rt + a. In both 

 cases H is zero outside the limits named. 



