34] 



Therefore this duality does not attain expression, not even in the s} stem 

 of Wilson and Lewis, thongh the}' nse miils of the kind e^, Gi.e,, e,. 



The foregoing table (subjoined p. 340) presents a siiinmarj of 

 the products used by various authors. 



The table has been arranged dualistically. Each product has been 

 indicated by an example. For the multiplications we used in tlie 

 columns 1 and 3 the author's own notation, bnt for the quantities 

 we used all through the notations adopted in this paper. The dual 

 bivector only has been written with the customary asterisk, while 

 the commutative scalar of Wilson and Lewis has been indicated 

 by k. The products marked with *) do not correspond exactly to 

 the other systems, because these systems do not contain the non- 

 commutative scalar I. 



The system R^ contains the existing fragments and all the 

 existing multiplications and i-ules, and owing to the free rules of 

 calculation (21 and 25) it is eminently suited for practical purposes. 



Tke system El and the elliptic and hyperbolic geometry in three 

 dimensions. 



With a homogeneous interpretation of the fundamental variables 

 Bl corresponds to a projective geometry in three dimensions, a non 

 degenei-ated quadratic surface being invariant. If the nuits are 

 selected according to (16) the equation of the absolute surface in 

 point- resp. plane-coordinates is: 



•h' + ■<' t '^■.* + ^'Z = ^ 



and the geometry is elliptic. If, on the other hand the units are 

 selected according to (17) the geometry is hyperbolic. The free rules 

 of the system are the same foi' botli cases. To a fundamental 

 element a point with a number-value corresponds, to a quantity of 

 the second degree a sum of linear elements (Dyname) and to a 

 quantity of the third degree a planar element. The sub-system of 

 the quantities of the .second by-degree is a form of biquatei-nions, 

 which was first mentioned by Clifford') as a system of linear 

 elements in a non-euclidic three-dimensional space. Hence the 

 system lil t!ompleles these biquaternions to a system which also 

 contains points and planar elements. 



1) Preliminary sketch of biquaternions. Froc. Lend. Math. See. 4 (73) 381— 395; 

 Further notes on biquaternions. (JoU. Math. Papers (76) 385, 395. 



