348 Professor Hamitton on Conjugate Functions, 
denominators and divisors are to be supposed different from zero unless the contrary 
be mentioned expressly ; or that we shall never suwb-multiple nor divide by a null num- 
ber without expressly recording that we do so. 
On the Comparison of any one effective Step with any other, in the way of Ratio, 
and the Generation of any one such step from any other, in the way of Multipli- 
cation ; and on the Addition, Subtraction, Multiplication, and Division of Alge- 
braic Numbers in general, considered thus as Ratios or as Multipliers of Steps. 
é 
21. The foregoing remarks upon fractions lead naturally to the more general con- 
ception of algebraic ratio, as a complex relation of any one effective step to any 
other, determined by their relative largeness and relative direction ; and to a simi- 
larly extended conception of algebraic multiplication, as an act (of thought) which 
enlarges, or preserves, or diminishes the magnitude, while it preserves or reverses the 
direction, of any effective step proposed. In conformity with these conceptions, and 
by analogy to our former notations, if we denote by a and b any two effective steps, 
of which a may be called the antecedent or the multipiicand, and » the consequent 
or the product, we may employ the symbol » to denote the ratio of the consequent » 
to the antecedent a, or the algebraic number or multiplier by which we are to mul- 
tiply a as a multiplicand in order to generate » as a product: and if we still employ 
the mark of multiplication x, we may now write, in general, 
b 
b= = 
- a 
San: (163.) 
or, more concisely, 
pe ae ry ee a, (164.) 
that is, if we employ, for abridgement, a simple symbol, such as the italic letter a, to 
denote the same ratio or multiplier which is more fully denoted by the complex 
symbol _ 
It is an immediate consequence of these conceptions and definitions, that the fol- 
lowing relation holds good, 
aes Ai geees Teh (165.) 
a denoting any effective step, and » and v denoting any positive or contra-positive 
