Quaternions to the Theory of Relativity. 447 



Thus from (xvi.) by applying the transformation from 

 S' to S, i. e., writing — v instead of v in the formula 



^^r=W)^-{ r ' +v(1 - /3)SrV -^ vS,lV }' 



or writing u/= — vv and after a simple modification 



P=S{*'+*Sr'v(l-J)j. • • . (xxi.) 



It is, ol course, natural to measure from the instantaneous 

 position of the moving charge e, as it is viewed by observers 

 at rest in S. It is easy to take this new point of reference. 



For let the instant in S' be zero, i. e., t' = 0. From (i.) 



r ' = - v0Yt + r + v(l — /3) Srv, 

 and from (ii.) 



These are merely the Lorentz-Einstein formulae. 

 If t' = 0, 



r^r-f vSrv 



I 1 - 1) < xxii -> 



If is the initial position of the electron, i. <?., its position 

 -at time £ = 0, in the interval — i-Srv it will have moved to e 

 where Oe is — i? 2 vSrv. 



Fur. 1. 



v 2 v. Srv e 



We require our formula in terms of R and possibly the 

 angle T'eQ ; OQ is the direction of motion of the electron. 

 On substitution for r' in (xxi.) 



p = (r + ^vSrv)J = ^.E, . . (xxiii.) 



as is easily seen from the figure. 



