332 Professor Hamitton on Conjugate Functions, 
Again, we saw that if a factor » be null, the product is then null also, 
52 Oi 0) 2 (116.) 
because the multiples of a null multiple step are all themselves null steps. But if, in 
a product of two whole numbers, v x «, the first factor » (with which by (114.) 
the second factor v may be interchanged) be given, and effective, that is, if it be 
any given positive or contra-positive whole number, («=E0,) then its several multi- 
ples, or the products of the form v x », form an indefinite series of whole numbers, 
sem OlO Xius 2 One lV OPXiis) Onxcursnil -Xqus 2S. Us OOS lyases (117.) 
such that any proposed whole number w, whatever, must be cither a number of this 
series, or else included between two successive numbers of it, such as » x » and 
(1+v) x », being on the positive side of one of them, and on the contra-positive side 
of the other, in the complete series of whole numbers (103.). In the one case, we can 
satisfy the equation 
wv Xun, or, 9 Xn) +w=0, (118.) 
by a suitable choice of the whole number v; in the other case, we cannot indeed do 
this, but we can choose a whole number v, such that 
w=pt+(vxp), Or, O(vxp) +o=P~, (119.) 
p being a whole number which lies between 0 and » in the general series of whole 
numbers (103.), and which therefore has a quotity less than the quotity of that given 
first factor », and is positive or contra-positive according as « is positive or con- 
tra-positive. In each case, we may be said (by analogy to arithmetical division) to 
have algebraically divided (or rather measured), accurately or approximately, the 
whole number w by the whole number p, and to have found a whole number v which 
is either the accurate quotient (or measure), as in the case (118.), or else the next 
preceding integer, as in the other case (119.); in which last case the whole number p 
may be called the remainder of the division (or of the measuring). In this second 
case, namely, when it is impossible to perform the division, or the measuring, exactly, 
in whole numbers, because the proposed dividend, or mensurand, w, is not contained 
among the series (117.) of multiples of the proposed divisor, or measurer, », we may 
choose to consider as the approximate integer quotient, or measure, the next suc- 
ceeding whole number 1 + v, instead of the next preceding whole number v; and then 
we shall have a different remainder, 94+ p, such that 
w=(On+0)+(+v x p), (120.) 
