180 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
PART II. 
APPLICATIONS OF THE SYMBOLIC FORMS. 
I. Geometrical Applications of the Symbolic Forms. 
The geometrical applications which may be made of the principles and notation 
just explained are of great variety and importance; but, as I am anxious to dwell 
chiefly on physical applications, I shall only touch on this part of the subject. 
(59.) Surface symhoUcally considered. — In Symbolic Geometry the symbols of lines 
represent, not merely length but also direction. A line is supposed to be the geome- 
trical effect of the motion of a tracing-point, and the equivalence of lines is con- 
sidered altogether and exclusively with reference to the change which Fig- 25. 
they represent in the position of the tracing-point. We consider that 
AB is equivalent to A'B', when the two lines are of equal length, 
parallel, and traced the same way ; for then they represent equiva- 
lent changes of position of their respective tracing-points. It is usual to employ the 
notation AB to denote the line AB traced from A to B, and BA to denote the same 
line traced from B to A; and thus AB=A'B', but BA— —A'B'. It is clear then that 
equivalent lines must be, not only of equal length and parallel, but also must be 
traced the ^ame way. Again, the equivalence being considered only with reference 
to the tracing-point’s change of position, AC+CB is equivalent to A'B'. 
Now, following this analogy, and regarding surface as the effect of the motion of 
a tracing-line, just as a line is the effect of the motion of a tracing-point, we may 
employ symbols to denote surfaces, not merely as regards their 
numerical area, but also with reference to the manner in which they 
are generated by their tracing-lines; and we may also define the 
equivalence of surfaces altogether and exclusively with referenee to 
the change which they represent in the position of the tracing-line. 
Thus, if ABA'B', BCB'C', ACA'C' be three parallelograms, and if we 
conceive them to be generated, respectively, by the translatio7is of A'A along A'B', 
B'B along B'C', and A'A along A'C' ; it is clear that 
(ABA'B')-]- (BCB'C') is equivalent to (ACA'C'), 
just in the same sense that 
(AB-l-BC) is equivalent to (AC). 
Again, if ABCD and A'B'C'D' be two parallelograms, AB and 
AD being respectively parallel and equal to A'B' and A'D', we may 
consider these parallelograms as equivalent to each other, in the 
same sense exactly that the lines AB and A'B' are regarded as 
equivalent. Only, just as it is necessary for the equivalenee of 
AB and A'B', that they should be traced the same way by their re- 
spective tracing-pomts, so it is necessary to the equivalence of ABCD and A'B'C'D', 
that they should be traced the same way by their respective tracing-lities. Now here 
it is to be observed that a tracing-point is devoid of two important properties which 
Fig. 26. 
Fig. 27. 
C 
A' 
A 
