350 Professor HamiLton on Conjugate Functions, 
and shall suppose that similar definitions are established for the algebraical sums and 
products of more than two ratios, or general algebraic numbers. It follows that 
b!  b_ bi+b 
ats 4 
b! bb! bb __ bbb 171. 
a a renee i Cm 
&e. 
and that 
b’ b_bi 
bts ase 
b’! bio op bl! (172.) 
tel a ole 
A ratio between any two effective steps may be said to be positive or contra-positive, 
according as those two steps are co-directional or contra-directional, that is, according 
as their directions agree or differ; and then the product of any two or more positive 
or contra-positive ratios will evidently be contra-positive or positive according as there 
are or are not an odd number of contra-positive ratios, as factors of this product ; 
because the direction of a step is not altered or is restored, if it either be not reversed 
at all, or be reversed an even number of times. , 
Again, we may say, as in the case of fractions, that we subiract a ratio when we 
add its opposite, and that we divide by a ratio when we multiply by its reciprocal, if we 
agree to say that two ratios or numbers are opposites when they generate opposite 
steps by multiplication from one common step as a multiplicand, and if we call them 
reciprocals when their corresponding acts of multiplication are opposite acts, which 
destroy, each, the effect of the other ; and we may mark such opposites and reci- 
procals, by writing, as in the notation of fractions, 
aes eb when 2x = e(2x c) ; (173.) 
and 
bY 
a! 
fae when © x (2x c) Se: (174.) 
definitions from which it follows that 
and that 
