42 Mr. 0. Hecaviside on Maxwell's Electromagnetic 

 By equating coefScients of powers of z^ in (45) we get 



except with r=0; then 



To prove (45), expand the q function in powers of c7^. 

 Thus, symbolically written, 



sinhg^ ^^^^,p-i/ sin7n^A ^ ^ ^ ^ u'j) 

 q \ mv y' ' * ' * 



the operand being in the brackets, and p~^ meaning integra- 

 tion from to ^ with respect to t. Thus, in full, 



}, C sinhoi , 2r"' rsmmvt a^ C* sin mvt ,, 



-Icoswz^; i-rfm=-l cos ms h -tt i i «^ 



rj q ttJo L mv 2 J„ mv 



'+...]im. . (48) 





^ , sin mvt 

 tat — 



Now the value of the first term on the right is 

 1 2 



-5 -, or 0, 



V V 



when z is < , = , or >vf. 



Thus, in (48) if ^;>v^, since first term vanishes, so do all the 

 rest, because their values are deduced from that of the first by 

 integrations to t, which during the integrations is always <z/v. 

 Therefore the value of the left number of (45) is zero when 

 z>vt. In another form, disturbances cannot travel faster 

 than at speed v. 



But when z<vt in (48), it is clear that whilst t' goes from 

 to if or from to z/v, and then from z/v to t, the first integral 

 is zero from to z/v, so that the part z/v to t only counts. 

 Therefore the second term is 



2 a; 

 TT 2 



r rCUsinmvt ^1. ^(^^C*-,^" sin mvt , 



1 cos mz I at dm= - 77 i tat\ cos mz dm 



J LJ« mv A IT z }z \ mv 



V 



The third is, similarly, 



V 2^ 4 jz \ v^J V 2^42 V vV 



