Impulsive Motion of Electrified Systems. 121 



electric displacement across unit length of a line of latitude. 

 When E and e are positive they are directed away from the 

 pole X. 



Now consider the spherical cap formed by the revolution 

 of the arc PX (fig. 1) about OX. The flux of displacement 

 out from this cap by its edge and by its convex surface is 

 (Ke/lir). 2Trrsm0 + iq{l'-cos0). 



By (1) the flux of displacement into the cap by its concave 

 surface is 



(K/1tt)( *E*.27rr 2 sin0 6?0, 



*- o v 



or , „ „ % f e sm0tW 



or g(l + }i)(l — cos0) 



2(L— ncos-ff) ' 



Equating the inward to the outward flux, we find 



K^sin^^-Ai^-iJd-cose), 



and hence 



q sin 6 _ gu sin 6 , 



e ~ Kr(l- n co~s0) ~ Kr(i?- u cos 0) ' " ' w 



This result is of great importance since all the formulae for 

 the radiation arising from the impulsive or gradual change 

 of velocity of an electrified system can be deduced from it*. 



Since e is at right angles to the plane of r and u we can 

 express it in the vector form 



Kr v — uv 1 Kr r — xly 1 ^ ' 



Here heavy type denotes vectors, and I*! is a unit vector in 

 the direction of r. Also ur 1 is a scalar and Yr x u a vector 

 product. 



§4. In the expanding spherical wave the disturbance is 

 propagated at right angles to the electric force with velocity 

 r, and hence the associated magnetic force, which is at right 

 angles both to the electric force and to the direction of pro- 

 pagation, is given by the equation 



H=rKE, 



which holds in every case of propagation of electromagnetic 

 waves. 



* This result was first given by Dr. Heaviside, l The Electrician,' 

 Oct. 11, 1901. 



