﻿Q nat ernionic Form of Relativity, 799 



quaternion which is transformed by (II.) or by (II. a) an 

 L-quaternion and an R-quaternion, respectively *. 



Now, Q 6 .[ ]Q, being the transformer of both L and L c , is 

 also the transformer of their sum and of their difference, 

 L e. also of the scalar and of the vector parts of the quaternion 

 L separately, say s = SL and A = Y L. Now, & being a 

 scalar, we have 



/ 



Q c sQ=sQ c Q = s, 



i. e. s is an invariant. Then 



A' = Q C AQ, 



and since Q, Q c . are unit quaternions, the tensor of A is 

 another invariant. 



Tints, the scalar of any L-quaternion and the tensor of its 

 vector are invariants, while the vector itself is transformed 

 into 



v£'=qsrrjQ (in.; 



Or use the form L = o-(cose-f a sin e), where a is the unit 

 of A. Then a cos e and a sin e are invariants and con- 

 sequently also a and e, so that another form of the last 

 theorem will be : — 



The tensor and the angle (or argument^) of any L-quaternion 

 are invariants, while its axis is transformed by Q c [ ]Q. 



In quite the same way it will be seen that &R is invariant 



and V22'=Q[Vi2]<&, .... (III. a) 



or in other words : — 



The tensor and the angle of any It-quaternion are in- 

 variants, while its axis is transformed by Q[ ]Q C . 



If we wish to return to the generating factors a c &c, we 

 can write the above properties : 



$a e 'b r = &aj>. ...... (15^ 



Vaa'b' = Q c [Va c h]Q, (16) 



and similarly 



Sa be =&ab c (15 a) 



Va!bJ=Q[Yab e }GU (16 a) 



But as a rule it is better to avoid any splitting of quater- 

 nions, if we are to expect simplicity and other advantages 

 from the use of quaternionic language. 



* L, E, being initials of left, right, may remind us of the position of 

 that of the two generating- factors which (as a c or b c ) has the subscript «-, 

 t. e. which is cov. q c . 



