OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
173 
(36.) All Units of Longitudinal Translation are equivalent to each other. For let 
a and a' denote any two directed units whatever ; then, as above, 
(a+a') X (a— a')=0 ; 
wherefore, since aXa'=a'Xa, we have 
aXa=a^Xa^ 
In illustration of this result the third suggesting instance, that of Mechanical fVorh, 
may be quoted; inasmuch as work is the effect of longitudinal translation, and all 
units of work are equivalent to each other, no matter in what directions the working 
forces act. 
(37.) All Units of Lateral Translation, in the same or in parallel planes, are equiva- 
lent to each other. — Let a, (3, a', j3' lie in the same plane, and let d denote the angle 
which a' makes with a, and therefore that also which /3' makes with (3 (art. 16): 
then (art. 17 ) 
cc'=oc cos ^+|8 sin ^ 
(3'=acos sin 5 
.•., observing that a. a=j3./3=0, and a.j3=— jS.a, 
we have 
a',/3'=a.a(cos^^+sin^ ^)=a.a. 
Hence ail units of lateral translation in the same, or in parallel planes, are equivalent 
to each other. 
In illustration of this result, the second suggesting instance, that of a Couple, may 
be quoted ; for all unit-couples in the same or in parallel planes are equivalent to 
each other. 
(38.) Directrix . — Hence, in expressing a unit of lateral translation, it is only neces- 
sary to specify a plane parallel to that in which the translation takes place ; or, what 
is better and immediately suggested by the theory of couples, it is only necessary to 
specify a line at right angles to the plane of translation. Such a line I shall, however, 
designate by the word directrix^' not axis ; because there is no idea of rotation in- 
volved in the present theory, translation being a kind of motion essentially different 
from rotation. I shall assume y to be the directrix of a./3 and of all units of lateral 
translation in planes at right angles to y ; and, generally, I shall define the directrix 
of any unit of lateral translation to be a directed unit at right angles to the plane of 
that translation. 
(39.) But, since a.(3=—(3.c6, it is necessary to distinguish posi- Fig- 21 . 
live from negative translations ; and this may be done by giving J 
an appropriate sign to the directrix. I shall therefore assume rlj j 
generally, that my is the directrix of ma.^, m being any number JHlI jjJll 
positive or negative. Thus —y will be the directrix of — a. (3, that ■ . 
is, of 3.c(. And, hence, I may adopt the following criterion of ^ 
