107 



t^^x^^f^z^^ ^, 



Hence if a;, ?/, z be the rectangular coordinates of a point in 

 space, the modulus of the right line drawn to it from the ori- 

 gin is a fourth proportional to the projections of this line upon 

 the axis of x and the planes of xy and xz. And the amplitudes 

 are the angles between the axis of x and these two last pro- 

 jections. 



The construction thus obtained for the product of two 

 right lines obviously coincides with Mr. Warren's in the case 



where 2; = 0. 



iiz 

 The nullity of the triplet x-\-iy-\-jz-\-ij "^ involves the 



three equations, a? = 0, «/ = 0, ^ = 0. 



But, however well these triplets fulfil the requisitions of 

 multiplication, we find they will not stand the test of addition. 

 The sum of two such triplets is not necessarily a triplet ; nor 

 can we add two of them together, unless they happen to have 

 a common amplitude. 



What has been here said may readily be extended, for we 

 might develope the expression e'*+/x+'^'^ + &=•, in which i,j, k, 

 &c. represent {n — 1) distinct square roots of negative unity, 

 into a series of terms, such as 



A + ?B -f Jc -}- Ad -{-... 



. ., CD , .BD .. BC 



AAA 



A 



and, conversely, we may reduce a multiplet 

 a -}- ib -{-jc + /{d + . . . 



., cd , . bd . . be 

 + ?k [-HI h ?; \- . , . 



I --^bcd 



