182 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
Again, as regards sign, there are four varieties of the form u.v, namely u.v, 
( — w).{— i;),both and — both — But there are only varieties 
of the form uxv when v becomes identical with u ; inasmuch as v must be the same 
as u in sign as well as in magnitude, and consequently the notation does not admit of 
the variations (— w) Xu and ux{—u). Now the other remaining variations are uxu 
and {—u) X {—u), and these have both the same sign. 
Hence (supposing that u and v are units), since units of lateral translation may 
differ from each other in two particulars only, namely, sign and plane of translation, 
and since units of longitudinal translation are incapable of differing in these parti- 
culars, we may see the interpretation of the result in art. 3/, and its perfect consist- 
ency with that in art. 36. 
( 70 .) Symbol of a line drawn from a, given Point. — If we assume that simple letters, 
such as u, V, w always denote lines of particular lengths and drawn in particular 
directions, hut all starting from the Origin of Coordinates O ; then Fig. 29. 
the proper symbol for denoting a line v drawn from the point P, OP 
being u, will be 
v-{-u.v-\-uxv ; 
for V denotes the line v drawn from O, and u.v-\-uxv the lateral and 
longitudinal effects of translating it from O to P, which effects, as 
above stated, have reference only to the change of position of P. I shall reserve the 
consideration of this symbolization for a future occasion, as a striking instance of the 
same thing will be given in the next section. 
II. Statical Application of the Symbolic Forms. 
(71.) Equivalence of Parallel and Equal Translations. — As a necessary preliminary 
the Fundamental Assumption in art. 29 must be justified. To do this it is only neces- 
sary to bear in mind that all statical problems are reducible to the case of balancing 
forces acting on the same rigid body. Now let AB and CD be 
any two parallel and equal lines in the same rigid body ; join A 
and D, C and B ; the intersection E bisecting the two joining 
lines ; and let P and Q be two equal parallel forces acting at a 
A and D. 
Then P and Q are equivalent to P-f-Q at E, and P-pQ at E 
is equivalent to P at B and Q at C. We may therefore translate 
P from A to B, provided we, at the same time, translate Q from D to C. Whence it 
follows that the translation of Q from C to D must be equivalent to the translation 
of P from A to B. Thus the fundamental assumption is justified. 
( 72 .) Representation of Forces by Lines. — It will be remembered that the general 
symbolic representation of forces by lines assumes the truth of the Parallelogram 
of Forces. As a matter of curiosity it may be asked, is it possible to apply this Sym- 
bolization of Translation to prove the Parallelogram of forces, without assumhig that 
Fig. 30. 
P 
