WILSON AND Li:\\ IS. — UKI.ATIVITY. 435 



third geometry intermediate between the Euehdean and the non- 

 EucHdean which we have discussed. In I'-ucHdean phme geometry 

 there is no hne fixed in rotation, in our non-KucHdean plane geometry 

 there are two fixed directions, in this new case there is just one. If we 

 were to investigate this geometry, we should find one .set of (parallel) 

 singular lines and one set of non-singular lines. Every non-singular 

 line may be rotated into any other. Angles about any point range 

 from — 00 to + 00 on each side of the singular Hne through that point. 

 The interval along any line intercepted between two singular lines is 

 equal to the interval along any other line thus intercepted. Every 

 non-singular line is perpendicular to the singular lines, as the singular 

 line is complementary to the singular plane through it. 



Some Algebraic Rules. 



32. We shall develop here a number of important relations be- 

 tween outer products, inner products, and complements which will be 

 of frequent use later. ^Slany of these relations hold in any number 

 of dimensions. We shall consider primarily a non-Euclidean space 

 in which one of a set of mutually perpendicular lines is a (5)-line, the 

 rest being (7)-lines. But except for occasional differences of sign, 

 the results are valid in a Euclidean space. 



In a space of 7i dimensions, the complement of a vector of dimension- 

 ality p is itself of dimensionality n — p. If a is a scalar and a is a 

 vector of any dimensionality, then from the associative law for scalar 

 factors, we have 



(aa)* = (aa)'ki2...n = (aki2...„)'a — a(a«ki2...n) = a*'a = aa*. (22) 



Let a, t% . . . be vectors of the respective dimensionalities p, 

 q . . . . Then 



/?xa = (— l)P?ax^9. (23) 



Owing to the availability of the distributive laws it is sufficient to 

 prove such relations as this for the simpler case where the constituent 

 vectors a, ^3 are unit vectors k^..., k/,... of dimensionality p, q. 

 In the permutation of a and j9, there are involved jjq simple transposi- 

 tions of subscripts; for each subscript in k;,... has to be carried 

 past all the subscripts of k^... Hence there are pq changes of sign. 

 Hence the outer product is commutative if either of the factors is 

 even, but is anti-commutative if both factors are odd in dimensiona- 

 litv. 



