OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
167 
My object here, however, in the remarks just made, is to point out the importance 
of distinctly marking" the two significations of the sign +. 
(14.) In all cases that I shall be concerned with, successive and simultaneous addi- 
tion are virtually equivalent, so far as the representation of directed magnitudes by 
lines is concerned. As regards the statical effect of forces, the Parallelogram of 
Forces shows this : as regards the dynamical effect, the Second Law of Motion (I 
mean Newton’s 2nd Law) does the same. As regards velocities and displacements 
the thing is obvious. 
(15.) Directed Units. I shall call an arrow of a unity of length (whether it represents 
a traced line, a force, a velocity, or any other kind of directed magnitude) a “ directed 
unit.” I shall always reserve the letters a, (3, 7 to denote a set of three directed 
units at right angles to each other. Hence, if AX, AY, AZ be 
three rectangular axes to which a, (3, 7 are respectively parallel ; 
and if x, y, z denote numerically the three coordinates of any 
point P ; xa, y^, zy will be the symbols representing these three 
coordinates in magnitude and direction, inasmuch as xa means 
X directed units put together by successive or geometrical addi- 
tion, all in the direction parallel to AX ; and so also as regards 
i/j3 and zy. Also if u be taken to denote the line AP in magni- ^ 
tude and direction, we have, by successive addition, 
u=^xa-\-y^-\-zy. 
The point P is often called the point (xyz), I may therefore speak of it as the point 
(u), inasmuch as u completely defines its position. 
If X, Y, Z denote numerically three forces parallel to AX, AY, AZ, it is clear that 
their complete symbolical representatives are Xa, Y/3, Z7. Also, if U denote the 
resultant of these three forces, we have 
U=Xa-l-Yi3+Z7. 
But here -f denotes simultaneous addition : we may, however, assuming the truth 
of the Parallelogram of Forces, regard it as the successive -J-, if we please. 
(16.) As just observed, I shall always suppose a, f3, 7 to be a set of three rectangular 
directed units ; I shall suppose the same also as regards a', j3', 7' ; a", (3", y", 8cc., using 
the dashes to denote different sets of directed units ; but the three in each set are 
always assumed to be at right angles to each other, unless the contrary be specified. 
In speaking of lines as regards magnitude and direction, I shall always use the 
word direction” as equivalent to “directed unit;” thus I shall call a the ^’■direc- 
tion ” of the line xa. The complete symbol of a line may be therefore described as its 
direction multiplied by its magnitude. 
(17-) If r denote the magnitude, and a' the direction of u, u=rci, and therefore, 
putting for u its value above, we find 
Fig. 10. 
or 
a'=aa-\-b^-\-cy. 
