340 Professor Hamiiton on Conjugate Functions, 
cals), when (though oz themselves the marks of opposite acts), they generate opposite 
Or ° os : 
steps, such as thesteps —” x » and - x b; and to mark this opposition we may write 
7 
== = (3) =. (137.) 
Hence every fractional number, with any positive or contra-positive whole numbers j 
and » for its denominator and numerator, may be put under one or other of the two 
following forms : 
n 
Ist, =, or IInd O-, (138.) 
m 
(m and n denoting positive whole numbers,) according as the proposed whole numbers 
u and v agree or differ in respect of being positive or contra-positive ; and in the 
Ist case we may say that the fractional number itself is positive, but in the IInd 
case that it is contra-positive : definitions which agree with and include the former 
conceptions of positive and contra-positive whole numbers, when we consider these as 
equivalent to fractional numbers in which the numerator is a multiple of the denomi- 
nator ; and lead us to regard the reciprocal of any positive or contra-positive whole 
number (and more generally the reciprocal of any positive or contra-positive frac- 
tional number) as positive or contra-positive like it; a fractional number being 
equivalent to the reciprocal of a whole number, when the denominator is a multiple 
of the numerator. A fraction of a late-making step b is itself a late-making or an 
early-making step, according as the multiplying fractional number is positive or 
contra-positive ; and as we have agreed to write b > 0 when »b is a late-making step, 
so we may now agree to write 
7 > 0, when a <b> Omandehs om Os (139.) 
. Uae Ong . . 
that is, when — is a positive fractional number, and to write, on the contrary, 
bh 
v 
i <o when ; xb<0O and v> 0, (140.) 
that is, when - is a contra-positive fractional number. More generally, we shall 
write 
f . v v 
, if =x b>-x b, b>9O, (141.) 
u lu 
and 
» if Dx bcix, b> Os (142.) 
