178 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
because the latter is indefinite, not distinguishing wliat particular root of — 
is meant ; but the former is perfectly definite, inasmuch as it indicates one particular 
root of — , namely, that root which denotes rotation through 90° about a as axis*. 
* Sir W. Hamilton, in his System of Symbolic Geometry and Quaternions, which may be truly described 
as one of the most profound and beautiful theories in the whole range of abstract science, assumes the letters 
i,j, k to denote particular values of V — In a certain limited sense, the symbols (Da.), (D^.), (Dy.) 
are equivalent to these ; for i,J, k denote rotation through 90° about the axes a, y; and thus W’e have 
y=i/3, a=Jy. ^=ka; 
and, therefore, since 
y=Da./3, a=D/3.y, j3=Dy.a, 
it is clear that i,j, k, and (Da.), (D/3.), (Dy.), so far, denote the same operations. 
But the operations are different in general-, for i^— — 1 always, but (Da.)®, as has been shown above, is not 
equivalent to —1, except when performed on lines at right angles to a. 
There is one difficulty, I confess, I cannot get over in Sir W. Hamilton’s Theory, no doubt from some 
misconception on my part, or from taking too narrow a view of the meaning of the sign ^ — 1 , 'The difficulty 
I allude to consists in this. Sir W. Hamilton assumes i, j, k not only to be particular values of — 1, but 
also absolute directions (i. e. units of direction) : in short he uses i,j, k in the same sense as a, /3, y above, and in 
the same sense also as (Da.), (D/3.), (Dy.). Now my difficulty arises from my not being able to see how 
particular values of — 1 can denote anything (geometrically) but change of direction, or to perceive, that 
they can be used with propriety as symbols of those rectangular units of direction. 
However this may be, it is important to explain the fact, that, in the results to which I have been led by 
the conception of translation, there are no general relations corresponding to 
— 1, y®= — 1, A:®= — 1. 
In my method, a, /3, y are simple units of direction and nothing more ; and instead of the relations just put 
down, I have been led, by the conception of translation, to the following, viz. — 
a.a=0, /3./3=0, y.y=0 
axa=l, /3x/3=l, yxy=l; 
though, as regards the last three relations, all that I have a right to assert as a matter of necessity is, that 
axa=/3 x/3=y Xy. 
Also, I find 
(Da.)®=-1, (D/3.)®=-l, (Dy.)®=-1. 
but only when performed on lines at right angles to a, /3, y respectively. 
I may observe that if uv denote the product of u and v according to Sir W. Hamilton’s Theory, it may be 
thus expressed in terms of my translation-products ; viz. — 
Mi;=— mXu + Dm.v. 
Hence, since Dm.i?=? — Dv.u, and « Xi;=i; x «, 
vu= — uxv—Du.v. 
Wherefore 
M X v= — —{uv-\-vu) 
Du .v==^{uv -^vu) . 
I may also observe, that, according to my method, I might put 
uv=-u 'X.v-\-u.v, 
supposing that uv denotes the complete product of the translation of u along v, including both the lateral and 
longitudinal effects. But I cannot make any nearer approximation to the equation 
uv=- — u X v + Dm.i; ; 
nor can I see that the conception of translation furnishes any interpretation of the — before a X r. 
If, in any way, I could show that — 1 was the proper value for a unit of longitudinal translation, I should have 
aa=axa + a.a= — I. 
