52 
1. If two couplets be multiplied together, the modulus of 
the product will be equal to the product of the moduli of the 
Jactors. 
2. The amplitude of the product will be equal to the sum 
of the amplitudes of the factors. 
One of the most important analytical properties of the 
couplet «+ VY —1.y consists in this, that the equation 
oe a sae y= 0 
is equivalent to two, viz. x = 0 and y = 0. 
As regards the geometric interpretation of the foregoing 
results, it is sufficient to observe that the symbol Y —1 has 
been explained as denoting rotation through a right angle; 
whilst the couplet 2 + V—1. has been taken to represent 
both the length and the direction of the right line drawn from 
the origin to the point whose rectangular coordinates are # and 
y: the length of this right line is obviously 7; and it is in- 
clined to the axis of x at an angle equal to 6. 
The problem now proposed by Mr. Graves is to assign two 
distributive symbols, , and «, of such a nature that (1) the sum 
or product of two triplets, e+ w—+x«z and 2+ uw, + «2, shall 
be itself a triplet of the same form: that (2) there shall be 
theorems concerning the moduli and amplitudes of triplets, 
similar to those already enunciated for couplets: that (3) the 
equation «+ .W-+«z=0 shall be equivalent to the three, 
x=0, y= 0, z= 0: and that (4) the symbols ¢ and « shall 
admit of a geometric interpretation analogous to that which has 
been provided for the symbol VY —1. 
The preceding conditions will be complied with, if we as- 
sume . and « to be distributive symbols of operation, which, 
when combined, are subject to the following laws: 
K(a) =a: Ki(a) = ai'«(a) = (a) *(e) eee 
We must, at the same time, agree to regard ¢(1) and «(1) as 
units absolutely differing in kind from each other and from 
