166 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEFflON 
is that given to + in Symbolical Mechanics, If u and v denote two Fig 
forces of certain magnitudes and directions, u-\-v is assumed to denote ^ 
u and V 'put together as in figure 7 ; that is, the origin of v coinciding, not y 
with the end-point, as before, but with the origin of u. The distinction I ^ 
allude to here is of considerable importance, and requires to be very closely attended 
to in applying lines to represent forces generally, as will appear. It might be well 
to distinguish these two significations of + by appropriate terms. I cannot think 
of any better words, for the purpose, than the two, “successive" and “simultaneous " 
the putting together in fig. 6 is manifestly effected by tracing the two lines in imme- 
diate succession, while that in fig. 7 is a simultaneous application at the same origin. 
I shall therefore call the putting together in fig. 6 successive addition, and that in fig. 7 
simultaneous addition. 
(12.) Signification of the sign =. Like +? the sign = denotes equivalence in a 
certain sense supposed to he understood ; thus in the example, 3£+55.=780c?., it denotes 
equivalence as regards the conventional value of certain coins. In Fig. 8. 
Symbolical Geometry = has reference to the change of position of the 
tracing-point by which lines are supposed to be drawn. Thus if u, v, w 
denote three traced lines, the equation, u-{-v=iw, means, that the 
tracing of u-\-v is the same thing as the tracing of w, so far as the change of position 
of the tracing-point is concerned. In this sense it is clear, that w must be the third 
side of the triangle in fig. 8. In Symbolical Mechanics = has reference Fig. 9 . 
to mechanical effect. Thus if u, v, w be three forces, the equation, 
u-\-v='w, means, that the mechanical effect of u-j-v is the same as that ^ 
w ; in other words, it means, that w is the resultant of u and v. 
(13.) Representationof Forces hy Lines. The suitability of lines to represent forces 
is obvious enough in ordinary Mechanics, where + has the signification of mere 
numerical addition ; but when we come to Symbolical Mechanics this suitability is 
no longer a thing to be assumed. A little consideration will show that the question, 
“ Can we assume lines to represent forces generally ? ” may be stated symbolically as 
follows, viz. If the lines u and v respectively represent the forces U and V, in mag- 
nitude and direction, will u-\-v also represent U-j-V in magnitude and direction? if 
not, the graphical mode of representation becomes inadmissible symbolically. Now, 
it is clear, by reference to figures 8 and 9, that this question amounts to asking, 
whether the Parallelogram of Forces is true or not ? for the peculiar signification of 
-|- in Symbolical Geometry makes u-\-v denote the diagonal of the paiallelogram 
constructed on u and v as sides ; whereas the Mechanical signification of -j- in U-f-V 
makes it denote the resultant of U and V. 
Hence it follows that the general representation of forces by lines assumes the 
truth of the Parallelogram of Forces as a necessary condition ; and, consequently, 
any symbolical proof of the Parallelogram of Forces which assumes that lines may 
be taken generally as representatives of forces, amounts to reasoning in a circle. 
