338 



ei .ei = + 1 



ei Xe23 = ei2.'5 = i Qa 



ei . ei2 = 62 



ei . 6234 = 1 



ei X 6123 = 62.3 



ei I = — I 6l =62.34 = - 



612 X 634 = I 



cycl. 1,2,3,4. 

 dual e — e 

 (compli- (22) 

 cated) 



lil =il 1= jl 



il Xji = ji Xii = I 

 V^ =-1 



I cycl. 



(24) 

 1,2,3. 



When applied to iiiiit« (lie rnlos for Ih and for e,, e,, e,, e, are: 



ei Xe2 = — 62 Xe =ei2 ei2 * 623 =61.3 



612 . 612 = — 1 

 612 . 6234 = ei3l 



612X6. 23 = 63 



e 12 1 = I 612 = — 634 



6123 X 62.34 = 614 



/ei 612.3 . 6123 = — 1 



6234 I = I 6234= 61 



1-2 = +1 



and for e^, e^, e,, e, : 



(See for formula (23) page 339). 

 The quantities of an even l)y-degree form a .sub-system with 8 

 units and the rules: 



jl * ]•-' =-J-' * jl =-i.3 



jl * i2 = i2 * jl =J3 I 



jl . jl -+1 



Ijl = jl I = + ii 



'il = 623 

 Jl = 601. 



But liiese are the same rules as those for the units e,, e,, 63, /e,, 

 /e,, /e, of R'i with ordinary complex coeflicients, so that the free 

 rules for A^'' also hold good for (piantities of an even by-degree 

 of RI if, instead of X ai^l • ^ve introduce the symbols * and 



X = .+ X: 



2a * 2b = quantity of the second by-degree 

 2a X 2b = scalar in I and 1 



2a X (2b * 2C) = 2a * 2b X 2C 



2a * (2b * 2C) = (2a X 2b) 2C — (2a X 2C) 2b 



2a (2b * 2C X 2d) = (2a X 2b) (2c * 2d) + ... . . (25) 



(2a * 2b) * (2C * 2d) = (2b X 2C) (2a * 2d) + . . . . 



(2a * 2b) X (2C * 2d) = (2b X 2C) (2a X 2d) + ... . 



(2a * 2b) (2c * 2d X 26) = (2b X 2C) (2a X 2d) 26 + ... . 

 (2a* 2b X C)(2d* 26 X 2f)=(2C X d)(2b X 26)(2a X 2f)+.... 



Hence these rules may be written down from memory, as well 

 as the others. 



The Sysiein Rl and the iheori/ of relativity {in an element of 

 four dimensional space). 

 Fragments of Rl have been used by various authors ') on the 

 theory of relativity. With them five products occur and two of these 



1) H. Minkowski, M. Abkaham, A. Sommerfeld, M. Laue, Ph. Fkank. 



