176 Mr. 0. Heaviside on the . 



The arbitrary quantities E b E 2 , . . . in (57) and (58), or 

 rather, as many of them as turn out to be independent, are 

 easily found to depend on the initial electromotive forces resi- 

 ding in those parts of the system in connexion with the cable, 

 either at the ends or intermediate, which influence v at time t 

 independently of its value /(a*) when £ = 0. 



If, for example, we join two points &\ and x 2 through a coil, 

 its self-induction will introduce one E ; and if this coil have 

 a closed circuit near it, a second independent E will be intro- 

 duced. 



25. Considering the line as of infinite length both ways, it 

 will be found that if 



v =f( x ) = SA sin (y + b\ . . . (59) 



where the a's are determined from 



. Jua — hoo? + h--.a° — ... , nr . s 



w= - l-vw-.. ■ » • • • ( 60 > 



then will v satisfy the differential equation 



( i+/i 4 + ^£ + ^£ + --)^ +? ) 

 = ( l _ A ^ + ^_^ + ...) /(a _ . (61) 



everywhere, thus expressing the relation between the values 

 of/(#) at any two points separated by a distance 21. Or, 

 which is the same thing, 



°-«s+v£+y£+.. v • • (62) 



where 



h= o + 4+h+h, 



12. It 



k - - + ^+ - 2 4-^4- 7, 4-7, 



15. it t £_ 



In the particular case 7^ = 0, h 2 = 0, ..., equation (61) 

 reduces to 



f( X + l)=f(x-l), (63) 



which simply expresses that f(x) is periodic, repeating itself 

 at intervals 21. 



Starting from this equation, or an equivalent one, Mr. 

 O'Kinealy (Phil. Mag. August 1874) proves Fourier's theo- 



