8344 Professor Hamitron on Conjugate Functions, 
(or whole) numbers a and #3, is not altered by adding any fractional (or whole) num- 
ber y to both, nor by subtracting it from both; so that 
> > 
y ee +a, and 0 y+P=0y + a, according as B=a. (151.) 
< < < 
19. Again, the composition and decomposition of swecessive acts of fractioning 
(instead of successive fractional steps) conduct to algebraical operations of multzpli- 
cation and division of fractional numbers, which are analogous to the arithmetical 
operations of multiplication and division of quotities. For if we first multiply a 
given step b by a given fractional number 4 that is, if we first perform on b the 
act of fractioning denoted by this number, aa so form the fractional step me x b, 
we may then perform on the result another act of fractioning denoted by aaather 
fractional number +, and so deduce another fractional step ~ x ( a ») ,which 
will evidently be itself a fraction of the original step b, and might therefore have 
been deduced from » by one compound act of fractioning ; and thus we may proceed 
to other and other fractions of that step, and to other compound acts of fractioning, 
which may be thus denoted, 
, Uy 
v v v v 7 
5x (= x b)=(4 x") x, 
Me BE ee 
" vihiog (152.) 
v v v 
mx dx(t xs) ba(5x = ) x bs Scen, 
B it # le 
, ah vy 
and the resultant fractional numbers ? x : , 4x4x", &c., which thus express 
fe 
the resultant acts of fractioning, derived on the proposed component acts marked 
by the fractional numbers 5 , =, 2 -,, &c., may be called the algebraic products 
of those proposed fractional caumbees and may be said to be formed by algebraically 
multiplying them as fractional factors together ; definitions which agree with the 
definitions of product and multiplication already established for whole numbers. 
The same definitions shew that every fraction may be regarded as the product of the 
numerator (as one factor) and the reciprocal of the denominator (as another); and 
give, in general, by (134.), the following rule for the calculation of a fractional 
product 
&e. (153.) 
