110 Dr. L. Silbersteiii : Farther Contributions 



distributive property of scalar multiplication of: vectors) 

 B x + M x, etc. Thus 



VA(B + M)=i|>i a (5 8 + J/ 3 )-A 3 (5 2 + A/o)]+etc, 



■ and this is the same tiling as 



VA(B + M) = VAB+VAM, . . . (19) 



•the distributive property of the vector product. 



This being valid, all other properties of such products and 

 of mixed products, AVBC, etc., known from ordinary vector 

 algebra, will continue to hold. It is therefore needless to 

 dwell upon them any further. It will be enough to notice 

 that the so-called " parallelepipedal " property, 



AVBC = BVCA = CVAB, .... (20) 



which is commonly proved by saying that each of these 

 three expressions represents the "volume" of the parallel e- 

 pipedon A, B, C (right-handed), will in our case be better 

 proved without the aid of the concept of " volume/' since 

 we have not defined any such concept. The vector YBC 

 being expanded as in (18), we see at once that its scalar 

 product into A is A X {BA 7 3 — B z ( \ 2 ) + etc, or 



A i A 2 A 3 

 Bi Bo B 3 



AVBC = 



( !, 0, 



and this is, by the well-known property of determinants, the 

 same thing as BVCA or CVAB. Q.E.I). 



The vector multiplication of two vector polynomials will 

 be handled as the scalar multiplication, the only difference 

 being that the order of factors must be preserved or, if 

 inverted, the sign of the pirtial product must be changed, 

 the well-known rule of ordinary vector algebra. 



8. General remarks. — Thus, having defined vector addition 



and vector multiplication with reference to a conventional 



7-line and a standard conic k (or 7-plane and quadric Q), 



we have seen that the resulting Vector Algebra and differ- 



. . . 



ential analysis are formally identical with ordinary, euclidean 



Vector Algebra and Analysis (the only difference being that 

 inches, etc. are replaced by staudthms, and that the angle 

 concept is modified and generalized). Does this mean that 

 we have involuntarily lapsed into euclidean space ? By no 

 means. We have only set up, in the general projective 

 space, a euclidean or a parabolic system of measurement. In 

 fact, that our vector algebra is entirely equivalent to para- 

 bolic metrics in the sense of Cayley-Klein, will be seen at 



