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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, 7 and j, denoting distinct 
distributive operations, such that 
#1) = 7°11) = — 1: #1) =je(1), 
and form with them and the three real magnitudes a, y, z the 
expression 
rpiytjo+ 7X. 
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 #(1) as an imaginary unit, differing both from 7(1) 
and j(1). 
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 
Z=Mcospcos'y, y= msingcosy, z= mcosPsiny; 
ve 
a 
m (cos cos x +7sin gd cosy +Jcos¢ siny +7ysingsiny), which 
is evidently equivalent to me’? tx, 
If then we call m the modulus, and ¢ and y 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 (a, y, z) are 
derived from its constituents by the equations 
the expression x+iy+jz +7 — will thus be transformed into 
