46] 
301 
46. 
NOTE ON A SYSTEM OF IMAGINAKIES. 
[From the Philosophical Magazine, vol. xxx. (1847), pp. 257—258.] 
The octuple system of imaginary quantities i u i 2 , i 3 , i 4 , %, i 6 , i 7 , which I mentioned 
in a former paper [21], (and the conditions for the combination of which are contained 
in the symbols 
123, 246, 374, 145, 275, 365, 167, 
i.e. in the formulae 
i 2 % 3 — i x , % 3 i x — i 2) ifa—z 3 , 
— 
^2^1 't'X 
with corresponding formulae for the other triplets i 2 i 4 i 6 , «fee.,) possesses the following 
property; namely, if i a , i p , i y be any three of the seven quantities which do not form 
a triplet, then 
(^aifi) • Iy ia ■ (iffty)' 
Thus, for instance, 
but 
(hi*) • is — h • % — ~ h 5 
i'i • (iiis) — h • ii ~ iii 
and similarly for any other such combination. When i a , ip, i y form a triplet, the two 
products are equal, and reduce themselves each to — 1, or each to +1, according to 
the order of the three quantities forming the triplet. Hence in the octuple system in 
question neither the commutative nor the distributive law holds, which is a still 
wider departure from the laws of ordinary algebra than that which is presented by 
Sir W. Hamilton’s quaternions. 
I may mention, that a system of coefficients, which I have obtained for the 
rectangular transformation of coordinates in n dimensions (Crelle, t. xxxii. [1846] “Sur 
quelques propriétés des Déterminans gauches” [52]), does not appear to be at all con 
nected with any system of imaginary quantities, though coinciding in the case of n = 3 
with those mentioned in my paper “ On Certain Results relating to Quaternions,” 
Phil. Mag. Feb. 1845, [20].