and on Algebra as the Science of Pure Tune. 331 
product, or (more fully) the algebraic product, of the whole numbers p» and v, whe- 
ther these, which we may call the fuctors of the product, be positive, or contra- 
positive, or null; and may speak of the process of combination of those numbers, as 
the multipling, or (more fully) the algebraic multipling of » by v: reserving still the 
more familiar arithmetical word ‘multiplying’? to be used in algebra in a more 
general sense, which includes the operation of multipling, and which there will soon 
be occasion to explain. 
In like manner, three or more whole numbers, », v, &, may be used successively to 
multiple a given step or one another, and so to generate a new derived multiple of the 
original step or number ; thus, we may write 
Ex fu x (ux a)}=é x {(v x #) x w= (Oa neat, (113.) 
the symbol ~ x v x » denoting here a new whole number, which may be called the 
algebraic product of the three whole numbers p, v, & those numbers themselves being 
called the factors of this product. With respect to the actual processes of such 
multipling, or the rules for forming such algebraic products of whole numbers, 
(whether positive, or contra-positive, or null,) it is sufficient to observe that the pro- 
duct is evidently null if any one of the factors be null, but that otherwise the product 
is contra-positive or positive, according as there is or is not an odd number (such as 
one, or three, or five, &c.) of contra-positive factors, because the direction of a step 
is not changed, or is restored, when it is either not reversed at all, or reversed an 
even number of times; and that, in every case, the quotity of the algebraic product 
is the arithmetical product of the quotities of the factors. Hence, by the properties 
of arithmetical products, or by the principles of the present essay, we see that in 
forming an algebraical product the order of the factors may be altered in any manner 
without altering the result, so that 
yXpopxr, ExvxpaspxeExv=Ke., &e. 5 (114) 
and that any one of the factors may be decomposed in any manner into algebraical 
parts or component whole numbers, according to the rules of algebraic addition and 
subtraction of whole numbers, and each part separately combined as a factor with the 
other factors to form a partial product, and then these partial products algebraically 
added together, and that the result will be the total product ; that is, 
(' #y) x n=O! xn) + (vxn), t (115.) 
yx (we t+p=( xv')4+ xp), &e. 
VOL. XVII. Sue 
