﻿704 Dr. L. Silberstein on the 



Hence ~ 



sin 2« = v^l— t* = 1/37 = ^^ 

 or 



2« = arctg(t/3; = aretg / L - \ . . . . (8) 



Now this is precisely the (imaginary) angle of rotation in the 

 plane t, x * of: Minkowski's four-dimensional world, corre- 

 sponding to the transformation (2). Hence, by (I.) and (7), 

 we may say that one half of this rotation is effected by Q 

 as a prefactor and the other half by the same quaternion as 

 a postfactor f. This circumstance throws a peculiar light 

 on each of our Q's. 



But what we are mainly concerned with is their union, 

 which considered as an operator may be written 



» = Q[]Q, ...... (La) 



the vacant place being destined for the operand. 



We have just seen that this simple operator converts the 

 quaternion q = r + cct into its relativistic correspondent q . 

 Our q is equivalent to Minkowski's " space-time-vector of 

 the first kind " or to Sommerfeld's " Vierervektor " x. y, ~, /. 

 These authors call by this same name any such and only such 

 tetrad of scalars (three real and the fourth imaginary) which 

 transforms in the same way as x, y. z, /, — adding where 

 it is necessary the emphasizing epithet " "Weltvector "J. 



* The axis of x coinciding with u, and x itself being our (ru). 



t At the first sight it might seem that, the axis of Q being u, this 

 quaternion turns r round u, i. e. in the plane ?/, z normal to u, while 

 in Minkowski's representation the rotation is in the plane x, t. But 

 this is only an apparent contradiction. In fact, 



Qr = cos a , r +sin a . Vur+ scalar, 



that is to say, Q as a prefactor turns the transversal component of r 

 round u by the angle -\-a and stretches its longitudinal component ; 

 similarly Q as a postfactor, besides stretching the longitudinal component 

 of r, turns its transversal component round u by the angle —a, thus 

 undoing the rotatory effect of the prefactor. Hence, what remains in the 

 final result is but a stretching of r's longitudinal component and a 

 change of / or t, and this amounts precisely to the Minkowskian rotation 

 in the plane x, t. 



% H.Minkowski, Die Grundghichungen filr d. elektromagn. Vorganqein 

 bewegten Korpem, Gotting. Nachrichten, 1908; Raum und Zeit, Physik. 

 Zeitschrift, vol. x. (1909), also separatim. A. Sommerfeld, Zur 

 Relativitat&theorie" i. and ii., Annalen d. Physik, vol. xxxii., xxxiii. (1910). 



See also the admirably clear and beautiful book Das Melatwitatsprineip 

 by M. Laue (Braunschweig, 1911), where the whole work of Einstein, 

 Minkowski, and Sommerfeld, together with the author\s own contributions, 

 will be found fully developed. 



