19-i Mr H. Cox, On the application of Quaternions. [Feb. 20, 



(2) On tJte application of Quaternions, and Grassmanns 

 Ausdehnungslehre to different kinds of uniform space. By 

 Homersham Cox, B.A., Fellow of Trinity College. 



The object of this paper is following Grassmann to establish 

 a pure algebraical calculus, the laws of which will coincide with 

 those of actual geometry. Such a geometry will differ from 

 ordinary algebra in having more than one independent unit. 

 Suppose A and B to be independent units such that there can be 

 no relation of the form A — xB when x is any ordinary algebraic 

 quantity real or imaginary. 



Then A and B are called points, and every expression of the 

 form p A + qB where p, q are numbers, positive, negative or 

 imaginary is called a point or some multiple of a point. pA +qB, 

 2 (pA + qB), 3 (pA + qB) are not considered different points, but 

 different multiples of the same point. 



Thus the point pA + qB varies with the ratio p : q and all the 

 points included in that expression form a singly infinite series 

 which is called a straight line. If A, B, C be distinct quantities 

 not connected by any linear relation, all the points included in the 

 expression pA+qB + rC form a doubly infinite series which is 

 called a line. This may be extended to three or more dimensions. 

 If P=xA + yB -f zC, {x,y,z) are said to be the homogeneous co- 

 ordinates of the point P, and it is shewn that Ix + my + nz = 

 is the condition P should, lie on a straight line. From these 

 definitions can be shewn to follow all the descriptive or projective 

 properties of curves considered as the loci of points satisfying 

 equations. 



It is found that there are three ways in which the idea of 

 distance may be introduced. If P, Q, R be single (not multiple) 

 points such that pP + qQ = rR and a = dist. PR, /3 = dist. RQ, 

 7 = dist. PQ so that a. + /3 = 7, then one of the following sets of 

 relations must hold 



ry 



r 2 =p 2 + q"+Zpq cosh j 



1 



P i 



p sinh j = q sinh ~ I 



q=P + q 



pa = q/3 



• J, 

 •II, 



