168 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
where a, h and c denote the cosines of the angles which a' makes with a, |3 and y 
respectively. 
If a' lies in the plane (a|3), and if 6 denote the angle which a' makes with a, this 
expression becomes 
a'=a cos ^+(3 sin 
(18.) According to the principles of Symbolical Algebra, we have 
l3=—icc, y=—i(5, ct=—iy, 
But it is to be remembered that the sign — ^ here does not denote the same identical 
operation in these three cases ; nor is it necessary that it should, any more than the 
sign — . The true state of the case is this, that (— ^)a is defined by the equation 
and the general solution of this equation is 
—ia=(3 cos d-\-y sin S, 
where & is perfectly arbitrary. Consequently the extraction of the square root of — 
gives, not simply two values, positive and negative, as in ordinary extractions of the 
square root, but an infinite number of values, namely the ivhole circle of directed 
units at right angles to a. 
I shall have no occasion to make any use of the sign —i, or any reference to the 
connections just given between a, and y, except in some future applications of my 
method, chiefly in Geometry. The statement just made is intended to show what 
a, (3, y are with reference to the square roots of — (or — 1), and to point out distinctly 
that a, |3, y are not supposed to be square roots of unity, but merely direction-units. 
(19.) Remarkable signification of da, d(3, dy. This signification I pointed out and 
made use of in a paper read before the Cambridge Philosophical Society (Nov. 1846), 
and it may be briefly stated here for the purpose of reference in certain applications 
of the present method. If a and os' be two directed-units at an in- 
definitely small angle to each other, we have (si—a — da% but a'— os is 
the line joining the extremities of os and ex!, and this line is at right 
angles to os and a' (ultimately), because a. and os' are lines of equal 
length. Hence da, is the expression for an indefinitely small line at right angles to a. 
This signification of da is one of great importance in Symbolical Geometry and 
Mechanics; thus for example, if os, (3, y denote direction-units fixed in a rigid body, 
the angular velocities of the rigid body are represented, in magnitude and direction, by 
da. dy 
dt’ df dV 
inasmuch as da, d^, dy represent, in magnitude and direction, the small angles 
described in the time t bv the extremities of these three direction-units. Of course 
I mean by the word “ angle f here, the circular arc which measures it. The im- 
portance of this signification of da, dj^, dy will be manifest in many parts of wdiat 
follows. 
