106 



therefore they are both equally fitted to serve as geometric 

 representations of the square root of negative unity. But 

 what is more, there is an infinite number of geometric ope- 

 rations of which this is equally true. For instance, rotation 

 through two right angles in any plane passing through the 

 axis of X would reverse the direction of a line placed upon that 

 axis. 



Let us take then two symbols, i and J, denoting distinct 

 distributive operations, such that 



z\l)=:f(l) = ~l:iji\)=ji(l), 



and form with them and the three real magnitudes x, y, z the 

 expression 



x + iy+jz + y^- 



As it depends upon three quantities, it may be looked upon as 

 a triplet; whilst it is, in some sense, a quadruplet, being 

 made up of units of four different kinds : for there is reason to 

 regard ^)'(l) as an imaginary unit, differing both from i{l) 

 andjXl). 



Before we proceed to consider the results arrived at in the 

 multiplication of such triplets, it will be convenient to change 

 their form. For this purpose let us put 



X zzm cos ^ cos\, y = msintpcosxt 2 zzmcos^sin^; 



the expression x + iy +J2 + y — will thus be transformed into 

 m (cos <f) cos x + i sin cos ^ +Jcos ^ sin ;i^ + y sin ^ sin ^), which 

 is evidently equivalent to jwe'*' +•'>'. 



If then we call m the modulus, and ^ and x the amplitudes 

 of the triplet, it will appear that the modulus of the product of 

 two triplets will be equal to the product of the moduli of the 

 factors : and each amplitude of the product will be equal to 

 the sum of the corresponding amplitudes in the factors. 



The modulus and amplitudes of the triplet {x, y, z) are 

 derived from its constituents by the equations 



i 



