and on Algebra as the Science of Pure Time. 339 
component acts of submultipling and multipling, without altering the final result, and 
may write 
p= = x(t x Gx hb): (183.) 
Iwi 
In general it may easily be shown, by pursuing a reasoning of the same sort, that in 
any set of acts of multipling and submultipling, to be performed successively on any 
one original step, the order of succession of those acts may be altered in any arbitrary 
manner, without altering the final result. We may therefore compound any proposed 
set of successive acts of fractioning, by compounding first the several acts of submul- 
tipling by the several denominators into the one act of submultipling by the product 
of those denominators, and then the several acts of multipling by the several numerators 
into the one act of multipling by the product of those numerators, and finally the two 
acts thus derived into one last resultant act of fractioning ; that is, we have the relations, 
v vy xv 
(= xb ear 
u CF ) wx pe d 9 
" U a yf. 134. 
{ex Exh = ee ys, Satie 
K le ro wx wx fe 
C. J 
We may also introduce or remove any positive or contra-positive whole number as a 
factor in both the numerator and the denominator of any fraction, without making 
any real alteration ; that is, the following relation holds good : 
= - (185.) 
whatever positive or contra-positive whole numbers may be denoted by pvw; a 
theorem which may often enable us to put a proposed fraction under a form more simple 
in itself, or more convenient for comparison with others. As particular cases of this 
theorem, corresponding to the case when the common factor w is contra-positive one, 
we have 
vy Ov Ov v 
that is, the denominator of any fraction may be changed from contra-positive to posi- 
tive, or from positive to contra-positive, without making any real change, provided 
that the numerator is also changed to its own opposite whole number. Two frac- 
tional numbers, such as 2” and P , may be said to be opposites, (though not recipro- 
fh 
VOL. XVII. Sie 
