330 Professor Hamiiton on Conjugate Functions, 
tive five, and the result is contra-positive three ; that is, the given step 2 x a or 2a 
x 
g 
may be decomposed into two others, of which the given component step 5 x a is one, 
and the sought compo: en’ step 3 Oa is the other. 
15. Any multiple step «a may be treated as a new base, or new unit-step; and 
thus we may generate from it a new system of multiple steps. It is evident that these » 
multiples of a multiple of a step are themselves also multiples of that step; that is, if 
we first multiple a given base or unit-step a by any whole number », and then again 
multiple the result « x a by any other whole number »v, the final result v x (u x a) 
will necessary be of the form w x a, w being another whole number. It is easy also 
to see that the new multipling number, such as w, of the new or derived multiple, must 
pend on the old or given multipling numbers, such as » and », and on those alone ; 
and the law ef its dependence on them may be conveniently expressed by the same 
mark of combination x which we have already used to combine any multipling 
number with its base ; so that we may agree to write 
w =v Xp, when» x a=v x (ux a). eyo) 
With this definition of the effect of the combining sign x, when interposed between 
the signs of two whole numbers, we may write 
yx (ue Xa) = (Xp) Kia x 25 (112.) 
omitting the parentheses as unnecessary ; because, although their absence permits us 
to interpret the complex symbol v x « x a either as v x (u x a) or as (vxp) Xa, 
yet both the processes of combination thus denoted conduct to one common result, or 
ultimate multiple step. . (Compare article 11.) 
When x and v are positive numbers, the law of combination expressed by the nota- 
tion » x #, as above explained, is easily seen to be that which is called Multiplication 
in elementary Arithmetic, namely, the arithmetical addition of a given number v of 
equal quotities » ; and the resulting quotity v x m is the arithmetical product of the 
numbers to be combined, or the product of « multiplied by v: thus we must, by the 
2 
definition (112.), interpret 3 x 2 as denoting the positive number 6, because 
3x (2x a)=6 x a, the triple of the double of any step a being the sextuple of that 
step; and the quotity 6 is, for the same reason, the arithmetical product of 2 multi- 
plied by 3, in the sense of being the answer to the question, How many things or 
thoughts (in this case, steps) are contained in a total group, if that total group be 
composed of 3 partial groups, and if 2 such things or thoughts be contained in_ 
each of these? From this analogy to arithmetic, we may in general call v x » the 
