342 Professor Hamitton on Conjugate Functions, 
“> or = or <.” If it had been merely required to prepare two fractional numbers 
so as to make them have a common denominator, without obliging that denominator 
to be positive, we might have done so ina simpler manner by the formula (135.), 
namely by multipling the numerator and denominator of each fraction by the deno- 
minator of the other fraction, that is, by employing the following expressions, 
te onthe: BES (147.) 
pte 7 
aX mo XE 
y 
uw 
a process which may be still farther simplified when the original denominators have 
any whole number (other than positive or contra-positive one) for a common factor, 
since it is sufficient then to multiple by the factors which are not thus common, that 
is, to employ the expressions, 
v xy v si vxp 
~ Ny (148.) 
ox wxexpe WX WXMXE 
A similar process of preparation applies to more fractions than two. 
18. This reduction of different fractional numbers to a common denominator is 
chiefly useful in combining them by certain operations which may be called algebraical 
addition and subtraction of fractions, (from their analogy to the algebraical addition 
and subtraction of whole numbers, considered in the 14th article, and to the arith- 
metical operations of addition and subtraction of quotities,) and which present them- 
selves in considering the composition and decomposition of fractional steps. For 
we compound, as successive steps, any two or more fractions ” x b, — x b, &c., of 
. . ine . 
any one effective step b, and generate thereby a new effective step, this resultant step 
will evidently be itself a fraction of the step b, which we may agree to denote as 
follows : 
(3 9) +( 
* x, 
(3 xo) + (5 xo) +(- 
and the resultant fractional number ~,+~ or 4+ Gare &c. may be called the 
Heo) 
fe (149.) 
oe) et) 3 &e.; 
w 
v v 
, Bes v 
algebraical swm of the proposed fractional numbers -, —, >, &c. and may 
. Hu He Oe . 
be said to be formed by algebraically adding them together ; definitions which agree 
with those established in the 14th article, when the fractional numbers reduce them- 
selves to whole numbers. If the denominators of the proposed fractions be the same, 
