338 Professor Hamitton on Conjugate Functions, 
that is, the whole number », regarded as a multiplier, or as a ratio, may be put under 
the fractional form © 
7? 80 that we may write 
eI 
7/3 (128.) 
and the reciprocal of this whole number, or the connected reciprocal number up» to 
multiply by which is equivalent to submultipling by p, coincides with the reciprocal frac- 
tional number 2 so that 
results which were indeed anticipated in the remarks made at the close of the fore- 
going article, respecting the extent of the conception of fractional numbers, as inelud- 
ing whole numbers and their reciprocals. As an example of these results, the double 
of any step a may be denoted by the symbol 7 x a as well as by 2 x a, and the half 
of that step a may be denoted either by the symbol 5 x a, orbyu2xa. The sym- 
bol u 1 is evidently equivalent to 1, the number positive one being its own reciprocal ; 
and the opposite number, contra-positive one, has the same property, because to re- 
verse the direction of a step is an act which destroys itself by repetition, leaving the 
last resulting step the same as the original; we have therefore the equations, 
wr ONZSOW: (180.) 
By the definition of a fraction, as a multiple of a submultiple, we may now express 
it as follows : 
v 
Y bau x(2 xb) = vx (up xd). (131.) 
ph 
Besides, under the conditions (122.), we have, by (112.) and (114.), that is, by the 
principle of the indifference of the order in which any two successive multiplings are 
performed, 
wxc=px (vx a)=(u xv) Sa) <M) Se ee DO (4X a)=v xb; (182.) 
so that a fractional product « = xb may be derived from the multiplicand b, by 
first multipling by the numerator v and then submultipling by the denominator p, in- 
stead of first submultipling by the latter and afterwards multipling by the former ; 
that is, in any act of fractioning, we may change the order of the two successive and 
, 
