174 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROxM THE CONCEPTION 
sign. I shall suppose myself standing at right angles to the plane of translation in 
such a position that the translated magnitude points to the right, while I face the 
direction in which the translation takes place ; and then I shall take an arrow 
pointing from foot to head as the directrix. According to this criterion the letters 
written underneath the following translations are their respective directrices ; viz. 
a.|3 |3.y y.a (3 . 05 y./3 a.y 
y a /3 — y — a — (3 
(40.) Measurement of a Translation. — Let u.v, or uXv, be the translation, a the 
direction of u, (aj3) the plane of u.v, 6 the angle which v makes with u, m and n the 
magnitudes of u and v ; then 
u=mtt,, 2 ;=w,(a cos sin ^) ; 
.'. u.v=mn^\\\.&{a.^), (since a.a=0). 
Hence there are mn sin 0 units in the translation w.^; ; mn sin d, therefore, is the nume- 
rical magnitude of u.v, and its directrix is (mwsin ^)y. 
Again, uXv=mn cos ^{ccX a), (since C4 x/3=0). Hence the numerical magnitude of 
M X is mn cos A 
(41.) It is worth remarking that mn sin 6, the numerical magnitude u.v, is the 
area of the parallelogram completed on u and v as sides. 
(42.) The Directrix of the Sum of two translations is the Sum of their Directrices . — 
Let the two translations be ma./3 and m!a'.^’, and their directrices, of course, my and 
m!y' ; let EB be the intersection of the two planes of these trans- Fig. 22 . 
lations, and take AB=w and BC=/«', AB and BC being 
drawn at right angles to BE, AB in the plane of a. (3, and BC 
in the plane of 
Now, since all units of lateral translation in the same plane 
are equivalent to each other, I may turn a and (3, a! and j3' 
about in their respective planes, until both a and a' coincide 
with BE ; in which case (3 and (3' will coincide with AB 
and BC respectively ; then AB will become m(3, and BC m'(3'. Let the third side of 
the triangle ABC be m"(3'' as shown in the figure. Then the sum of the two transla- 
tions is 
moi.(3-\-ni'ci'.(3', 
which, since u=a, becomes 
a.(mj8-l-r/i'/3') ; 
and this, since becomes 
m"(n.(3". 
Let m"y'' be the directrix of this : then it is clear that y, y' and y", being each at right 
angles to a, lie in the plane of the triangle ABC, and consequently my, m'y and 
m"y" are the three sides of the triangle ABC, supposing it to be turned round in its 
plane through 90°. It follows therefore that 
nd'y” = my -f- m'y’. 
