﻿798 Dr. L. Silberstein on the 



The reciprocal of a physical quaternion is also a physical 

 quaternion. For we have 



a -1 =sa c (T<2)~ 2 , 



while the tensor Ta of a physical quaternion is already 

 known to be an invariant. Notice that a and a~ x are 

 mutually antivariant. 



Now for the product of physical quaternions. Take any 

 pair a, b of such quaternions. Leave aside a b which trans- 

 forms in the unmanageable way a f b' = QaQ 2 bQ j (a. b being 

 torn asunder), and pass at once to the product of antivariant 

 factors, which might perhaps be called the alternating product, 

 say 



L = a c b (13) 



Then I/=Q c a c Q c . QhQ, whence by the associative property, 

 and remembering that Q C Q = 1, 



L' = Q,LQ (13') 



Thus, L is certainly not a physical quaternion of the kind 

 already considered ; but sincej it is transformed in such a 

 simple way and since it has, as will be seen in the sequel, an 

 almost immediate bearing upon relativistic Electromagnetism, 

 it deserves to be considered a little more fully. Consider, 

 then, the conjugate of L. Remember the elementary rule, 

 by which the conjugate of the product of any number of 

 quaternions is the product of their conjugates in the reversed 

 order, i. e. in our case 



L c =b c a (14) 



Now, transforming this, we get in quite the same way as 

 above 



L c ' = Q e L c Q (U') 



Hence we see that Q e [ ]Q (II.) 



is the relativistic transformer of both L = a c b and its conjugate 

 L e . Similarly, 



Q[]Q. (IT. a) 



will be the transformer of both R=«# c and its conjugate 

 R c = 6a t .. Thus the behaviour of L and R is characteristically 

 distinct from that of q or of q c . 



Without trying as yet to invent for these kinds of quater- 

 nions any particular names, let us provisionally call any 



