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

 The result is 

 v i^rij.« 3 - 5 ■ ■ l 2 -3 2 .5.7 , , 1 2 .3 2 .5 2 .7.9 



, a z' 3 3.5 2 , 1 2 .3 2 .5.7 \ 



+ rH + f Zl ~J\f 1 + |3|4 + ---J 



, l 2 3 2 a 2 ,/l ,3 3.5 , , P.3 2 .5.7 8 \ , n 



+ y^ Zi % i Zi ~m Zi ~^^ Zi -7 + -J' < 38l) 



the structure of which is sufficiently clear. Here z x = vtJSa. 

 This formula, when ^<a,holds between r=a and r— a— vt. 

 But when vt > a though < 2a, it holds between r=a and vt—a. 

 Except within the limits named, it is only a partial solution. 



70. As regards E 2 , it may be obtained from (381) by the 

 following changes. Change Ej to — E 2 on the left, and on 

 the right change Z\ to — z 2l where 



2 2 =z{vt + a— r)/Sa. 



It is therefore unnecessary to write out E 2 . This E 2 for- 

 mula will hold from r = a to r = vt + a, when vt< 2a ; but after 

 that, when the front of the return wave has passed r=a, it will 

 only hold between r=vt—a and vt + a. 



71. Next to find E 3 , the E in the cylinder when vt>a and 

 the solution is made up of two oppositely going waves, and 

 E 4 the external E after vt=2a, when it is made up of two 

 outward going waves. I have utterly failed to obtain intelli- 

 gible results by uniting the primary waves with a reflected 

 wave. But there is another method which is easier, and free 

 from the obscurity which attends the simultaneous use of U 

 and W. Thus, the equations of E 3 and E 4 are 



E 3 =-(7r/2^)W 0r W a ^, (382) 



E , = -(^ 3 /2)H^(^Jia)w.., . . (383) 



by (367) ; and a necessity of their validity is the presence of 

 two waves inside the cylinder, because of the use of J and J 1 ; 

 it is quite inadmissible to use J when only one wave is in 

 question, because Jor=l when r = 0, and being a differential 

 operator in rising powers of p, the meaning of (382) is that 

 we find E 3 at r by differentiations from E 3 at r = ; thus 

 (382) only begins to be valid when vt = a. 



To integrate (382), (383), it saves a little trouble to calcu- 

 late the time-integrals of E 3 and E 4 , say 



A 3 =-p-iE 3 , A^-^-%. . . . (384) 



