﻿804 Dr. L. Silberstein on the 



Now, these equations give immediately for the components 

 taken along u (the direction o£ motion) 



E/=B i; M^M,, 



and for the two other pairs of rectangular components (the 

 right-handed system being used) 



E 2 '= 7 (E 2 -/3M 3 ); M 2 '= 7 (M 2 +0B 3 ) 



E 3 '=7(E S +/3M;; ; M 3 '=y(M 3 -/3E 2 ), 



which are precisely the well-known transformational formula?, 

 obtained for the first time by Einstein. Thus (VII.) is 

 verified. 



Again, Q, Q c being unit-quaternions, we see from (VII.) 

 that, as already has been remarked, the tensor of F is an 

 invariant, 



TF' = TF, (VIII.) 



which may also be written, more conveniently*, F ,2 = F 2 . 

 Now, by (19), -F" 2 = M 2 -E 2 -2<EM) ; thus we see that 

 (VIII.) contains both of the well-known invariants of 

 Minkowski : 



M 2 -E 2 and (EM) (22) 



Notice that what is called a pure electromagnetic wave is 

 defined by M 2 = E 2 , (EM) =0. Using the above form we can 

 characterize a pure wave more simply by | 



TF = 0, or F 2 = FF = 0. 



Thus, by (VI II.), a wave which is pure to the S-inhabitants, 

 is also pure to the S'-mhabitants. But this example only by 

 the wav. 



Instead of the above F, as defined by (19), we may as well 

 take the complementary bivector 



G^M + tEj (19 a) 



Then we shall get as the quaternionic equivalent of the 

 electromagnetic equations (18), instead of and in quite the 

 same wav as (VI.), 



D C G=C C , (Yl.a) 



* Remember that, F being a scalarless quaternion, its conjugate is 

 simply — F. 



t This remark will be found also in my paper of 1907, cited above. 



\ G is a complex vector "conjugate " to F, in the sense of the word 

 used in the Theory of Functions. But to avoid confusion with the 

 quaternionic notion of conjugate, I do not call it by this name and do 

 not denote it by F 6 -. 



