OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
177 
Hence the operation (a(ax) performed on v gives the projection of v on u, and the 
operation (1 — (aXa) the coprojection ; where a denotes the “ direction ” of u, i. e. 
n u 
- or — =. 
^ V U X u 
(53.) Since «.?;=«. (AC +CB)=a.CB, it is clear that the symbol a.v denotes, in 
magnitude, the coprojection of on a ; and it also represents the plane of projection. 
But Tia.v is a better symbol to use ; for its magnitude is the same as that of a.v, and, 
in direction, it denotes a line at right angles to the plane of projection. 
(54.) Repetition of the Operation (Da.). This operation is often to be performed 
twice in investigations, and, on this account, the following relation is important, viz. 
(Da.)V=(aXa')a — a' , (1.) 
Of course (Da.)V means Da. (Da. a'). 
To prove this, let j3 be chosen so as to lie in the plane (aa'), and let 6 denote the 
angle which a and a' make with each other ; then (by art. 17) 
a' = a cos ^+|3 sin 6 
Da.a'=7sin^ (art. 48) 
Da. (Da. a') = — /3 cos 0 
= a cos d—a' 
or (Doi.ya-={aXoc')a, — a! (art. 50). 
\i u=ma v—na', we have 
{Tiu .yv=nfn{'Da fa' 
= {maxnc^)ma — {maX'>na)na' , 
or (Du.yv={uXv)u—{uXu)v\ (2.) 
or, ifm=l, {T)a.yv={aXv)a—v (3.) 
(55.) Relation of the Operation (Da.) to the operation \/ — 1 or ( — )^. The defini- 
tion of the index | in relation to operations is this. If H and be symbols of 
operation, such that H performed twice on a quantity gives the same result as 
once performed, then H is denoted by Clf Now, if we suppose that v is at right 
angles to a, and therefore axi^=0, the equation (3.) gives (what indeed is otherwise 
more easily shown from art. 48) 
(Da.yv= —v, 
wherefore 
Da.= (-)i 
In this case a is any unit line whatever at right angles to v, and therefore, in Solid 
Geometry, ( — )« or s/ —\ has not two (as in Plane Geometry) but an infinite 
number of values. 
(56). It is clear from the relation 
{T)a.yv=—v, 
that (Da.) has all the properties of the sign provided it he performed on lines 
at right angles to a. But (Da.) is a far better sign for actual use in Solid Geometry than 
MDCCCLTI. 2 A 
