OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
183 
forces are represented hy lines} It may be done very simply, and I give the proof 
here as an example of the application of the method. 
It may be seen, on referring to art. 21 and the following articles, that there is no 
assumption whatever of the possibility of representing forces by lines generally. The 
arrow representing the translated magnitude is used merely as a conventional symbol, 
just as a letter in algebra. I therefore may apply the notation u.v to the present 
question, provided I consider v to be the symbol of a force, and not of a line repre- 
senting that force. I must not however employ the reasoning in art. 32, for that 
distinctly assumes the point in question. The following then is the proof I shall give, 
(73.) Parallelogram of Forces . — Let A, B, C denote any Fig. 31 . 
three units of force, and a, h, c units of length (directed 
units) parallel respectively to A, B, C ; let X, Y, Z, x, y, % 
be pure numbers ; suppose that the forces XA, YB, ZC 
balance each other, and that xa, yh, zc are the three sides 
of a triangle formed by lines drawn parallel to the forces. Then by successive addi- 
tion we have 
xa-\-yb’\-zc= 0 , (1.) 
and by simultaneous addition 
XA-fYB+ZC=:0 (2.) 
Hence, from (1.) and (2.), we have 
(— 2c).(— ZC) = (.ra-l-3/Z>).(XA-fYB) ; 
or, observing that (since no lateral effect is produced by translating a magnitude in 
its own direction) c.C, a. A, 6.B are each zero, we have 
0=a?Y(a.B)-f3/X(6. A) ( 3 .) 
Now, without altering the directions of the units A, B, a, b, let us put X=Y ; in 
which case it is self-evident, that ZC must become equally inclined to XA and YB, 
and therefore zc must make equal angles with xa and 3/6, which gives x=y, Thus 
( 3 .) becomes 
a.B-l-&.A=0, or &.A= — a.B. 
Wherefore, restoring the inequality of X and Y, we find from ( 3 .), 
(xY— 3/X)a.B=0, or j:Y— 3/X=0. 
And similarly, we may show that 
yZ-zY^iQ 
z'X. — :rZ=0 ; 
whence X : Y \Z \ \ x-.y\z. 
And this is, virtually, the Parallelogram of Forces. 
Thus it appears that the notation m.z; is capable of affording a simple proof of the 
great fundamental theorem of Statics ; this application of the method is given, however, 
as I stated above, merely to show by example what can be done in this way. I may 
observe that the whole of the proof here given depends simply upon two things, the 
distributiveness of u.v, and the fact that numerical coefficients may be brought out 
