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

 or, expanding by (309), 



^iw^-^p?"-)^' (346) 



in which / may be any function of the time. Let it be zero 

 before and constant after £=0. Then, first, 



a*/o-/oO*r* (347) 



Next effect the integrations of this function indicated by the 

 inverse powers of q or p/v, thus 



*(*- 8 i +■■ •)(->-= ( wg) + i 4_(S- . . .)(-)-' 



= (l+««/2r)-*(7n70"*=(2r/7r)*[^(^ + 2r)]-*. . (348) 



Lastly, operating on this by e~^ r turns vt to vt—r, and brings 

 (346) to 



H = (/o/27rH(^ 2 -^ 2 )- 1 , . - • (349) 



which is ridiculously simple. Let Z be the time-integral of 

 H, then 



z =fi^[? + (?- ] W> • • ( 350 > 



from which we may derive E ; thus 



curl Z=C E, or-lL-lg- ^,^ (35 1) 

 The other vector-potential A, such that E = — ^>A is obviously 



A — sfK?-- 1 ) < 352 > 



All these formulae of course only commence when v£ reaches r. 

 The infinite values of E and H at the wave-front arise from 

 the infinite concentration of the curl of e at the axis. 

 Notice that 



E = H*/rc (353) 



everywhere. It follows from this connexion between E and 

 H (or from their full expressions) that 



cE 3 -^H 2 = ce 2 = c(/ /27rr) 2 ; . . . (354) 



where e denotes the intensity of impressed force at distance r, 

 when it is of the simplest type, above described. That is, the 



