% 
+ 
. 
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7 
TF eS a ee ae Ss. 
53 
the real unit. This, in fact, satisfies the third condition. As 
(a) = x(a) we may, for the future, dispense with the symbol 
«x, and write the triplet in the form «+.y+°z, or more shortly 
thus (2, y, z). 
In the first place, it is evident that the sum or product 
of two triplets is itself a triplet. 
Next, supposing (a, y, Z). (1, Y1, 21) = (2) Ya, Z2) we 
shall have the modulus of the product equal to the product of 
the moduli of the factors, if we call the expression 2? + y? + 
2° + (¢+e) (ay + yz + zx) the modulus of the triplet (a, y, z). 
And this modular theorem involves in it two others of the 
same kind, concerning the purely real moduli, « + y + z, and 
a? + y° +27 — xy —yz— za. According as we bring the 
triplet into different forms by changing the variables in it, 
there will be either two theorems relating to moduli, and one 
relating to amplitudes, or one modular theorem, and two con- 
cerning amplitudes. 
For the purpose of geometric interpretation let us suppose 
the three positive portions of the axes of rectangular coordi- 
nates to meet the surface of a sphere, whose centre is at the 
origin, in the points x, y, z, through which a small circle of 
the sphere is described, and let us give the name of symmetric 
axis to that diameter of the sphere, which passes through the 
poles of this circle. Now if we conceive the real unit placed 
on the axis of x, «(1) on the axis of y, and (1) on the axis of 
2, we may interpret the symbol : by saying that it denotes a 
conical rotation round the symmetric axis through an angle 
of 120 degrees. Three such operations, executed successively 
on the real unit, will bring it back to its original position 
on the axis of x. This is in accordance with the equations 
ux(@) = x(a) = (a) = a 
in virtue of which we may regard (1) as a purely imaginary 
cube root of positive unity. 
Mr. Graves mentioned that, since he had obtained per- 
