142 Dr. L. Silberstein on 



thus obtained separately for the components are but afterwards 

 combined into vector equations. 



Now the quaternionic method of treatment seems in this 

 respect also to be more, convenient and more transparent. 

 Tn fact, since we know already, by formula (XVI.), that 



Z=i{($n)-c™}+/n-^. . . (14) 



is a physical quaternion, cov. q, we have, by the same formula, 



QZQ=iG'*'F', (15) 



where, by (13) v 



k' = QkQ, say = «<r' + N' = w' + NV, 



./ =T [ o — /3(ni)]; N'=n[( 7 -l)(ni)-<7./3 7 ]i. (16) 



(Here i is written instead of the original u, so that now the 

 velocity of the system S' relatively to Sis v = ti = c/3i.) 

 Thus, by (15), 



QZQ=^W+^GVF'; 



— — 



but, by (XVI.a), (XVI.5), the right-hand side of this equation 

 is 



whereas QZQ equals, by (14), 



^l^-au-^-Zs^yj, +/n _ ?sp 



+i { (7-d(/;,- - e %)-fo(z% -™)} • 



Hence, by comparing separately the scalar and the vector 

 parts, we obtain the following two relations 



if-™)-eiu-°%)=^-«'u> . . . (i 7a) 



/n- f+i {(y-l)(/ al - ^y^f-au)] =fW- £* 



. . . (17 6) 

 Now, these being valid for any a (and for all directions of n), 



