448 Applications of Quaternions to Theory of Relativity. 



But 



p = E + VuH, 

 and since 



u = 0, p = E, 



E is the electrical intensity at P and the formula (xxiii.) 

 shows that it is directed along R, the line joining P to the 

 instantaneous position of the electron. 



From (xx.) since H/ = the magnetic intensity due to the 

 charge is 



H=— vVEv=yVvE. 



This immediately shows that H is perpendicular to v and E r 

 and in such a direction that a right-handed screw placed 

 along the direction would rotate v into E. 



It is immediately seen from the figure that the magnitude 

 of H is vE sin \ ± . 



It remains to put r' z in terms of K, these quantities 

 denoting the magnitudes OP' and eV. 



X denotes the angle between r and the direction v. Thus 



r' = r— vr cosX 1 1— -~V. 



Thus 



r' = AC. 



* 

 It immediately follows that 



r" = rXl-v 2 cos 2 \). 



Again from fig. 1 



R 2 = r 2 (l-2u 2 cos 2 A + v 4 cos 2 X) 



and R r 



sin A sin \' ' 



From these equations we derive 



r'^^R^l-^sin 2 ^)- 



