72 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Linear distance in hypcrspace. 



22. The argument by which we derived the formula for the distance 

 between two points in three dimensions apphes without change to 

 higher dimensions. The formula for distance is then always 



-—^JqAB)_ 



{4> A) (0 B)' 



where q- is a complex matrix of order n — 2 and ^ a hyperplane. Sim- 

 ilarly the angle between two hyperplanes is 



where p is a complex matrix of order two and F a point. We wish 

 these quantities to be invariant under the same group of collineations. 

 This will happen if and q are determined by F and p and conversely. 

 We shall therefore consider the system of complexes determined by 

 a point F and a complex p of the second order. The details of this 

 discussion depend somewhat on whether the space is of even or odd 

 order. We consequently consider these cases separately. 



23. Space of order n = 2 ???. The progressive products of a 

 complex p with itself give a system of complexes [p p], [p p p] etc. 

 we shall denote these by the symbols p~, p^ etc. In the present case 

 p^- is represented by a sum of determinants of order n and hence is a 

 scalar. We assume that this quantity is not zero. Such for example 

 is the case if 



V — Ai A.2 + ^3 .I4 + ^2m— 1 A.2m 



and the points Ai do not lie in a hyperplane. For then 



p"' = ml Ui^2 Aom). 



Since p"* is not zero none of the lower powers are zero. 

 We take as a fundamental system the quantities 



F, p,'Fp, f^ Fp"^-\ 



consisting of the powers of p and those powers multiplied by F. W^e 

 shall find that this system forms a group under progressive and regres- 

 sive multiplication, in the sense that the product of any two is either 

 zero or a numerical multiple of a third in the system. 



To form products it is sufficient to recall that ^ is a sum of products 

 of two points and hence in linear (distributive) operations behaves like 

 a simple product of two points. Furthermore to multiply regressively 



