and its application to the Geometry of three Dimensions. VIS 



amount of such rotation, and so may be considered equal to 

 5^(1+5), where jo is any real quantity whatsoever. There is 

 therefore no difference in either magnitude or direction be- 

 tween the lines 1 +5(1) and ^(1 +5(1)) ; and we are entitled to 

 put 



(I-5(l))(l+5(l))=0. 



8. I think the following will be found to be the true theory 

 of the vanishing of factors and products. 



In a system of algebra in which there is but one real mo- 

 dulus of multiplication, a factor and its modulusvanish together. 

 The vanishing of a factor will cause the vanishing of a product 

 into which it enters along with other finite factors; and con- 

 versely, the vanishing of a product indicates the evanescence 

 of one of the factors. This rule applies to the ordinary double 

 algebra, whose units are 1 and (—1)^, and also to Sir William 

 Hamilton's quaternion algebra of four units, 1, 2,7, k. 



But systems of algebra may be constructed in which the 

 case is different. The process of operating with one mixed 

 quantity upon another — that process, in fact, which, on the 

 score of analogy, seems to claim the title of multiplication — 

 may lead us to regard a mixed quantity as having more than 

 one real modulus: and these moduli, suppose n in numberj 

 need not all vanish simultaneously. When they do, the 

 quantity to which they belong vanishes likewise ; but we can- 

 not say that it does so unless its different moduli of multipli- 

 cation are separately equal to zero. Such a factor entering, 

 along with other finite factors, into a product, annihilates it 

 by annihilating all the « real moduli of the product. On the 

 other hand, the product is not reduced to zero unless all its 

 real moduli vanish ; and this cannot take place unless, amongst 

 all the moduli of all the factors, there be w, of different Icinds^ 

 separately equal to zero. These n vanishing moduli may in 

 general be distributed amongst the factors in various ways 

 without annihilating any one of them. What has been last 

 said applies to the systems of double and triple algebra dis- 

 cussed in the present paper. 



Operating with x + sy upon x' -{-s^, we found that the mixed 

 quantity x + s{y) had two real moduli of multiplication, <r+j/ 

 and J^— ^. It vanishes if both these are equal to 0, but not 

 otherwise. And x" + s{y")f the product of these two factors, 

 will not vanish unless both its real moduli likewise vanish. 

 Now it appears from equations (I.) that this may happen in 

 four different ways. 



1. We may have x=0 and y — 0; 2, x' = and y=0; 

 3, a?+^=0 and x'—i/=Q\ 4?, x—y—O and «'+y=0. In 



