326 Professor Hamitron on Conjugate Functions, 
It is manifest that in this notation 
nmOxa=in xOa=O(n ee (105.) 
and m x a=nOxO@a=O(nOxa)=O( nm xa), 
if 7» denote any one of the positive numbers 1, 2, 3, &c. and if m © denote the cor- 
responding contra-positive number, 1 0, 20, 30, &c.; for example, the equation 
“2x a= 2x @a is true, because it expresses that the second contra-positive multiple 
of the base a is the same step as the second positive multiple of the opposite base or 
step © a, the latter multiple being derived from this opposite base by merely doubling 
its length without reversing its direction, while the former is derived from the original 
base a itself by both reversing it in direction and doubling it in length, so that both 
processes conduct to the one common compound step, 02+ Oa. In like manner 
the equation 2 x a= 2 © x Oa is true, because by first reversing the direction of the 
original step a, and then taking the reversed step Oa as a new base, and forming the 
second contra-positive multiple of it, which is done by reversing and doubling, and 
which is the process of generation expressed by the symbol 20x Oa, we form in 
the end the same compound step, a +a, asif we had merely doubled a. We may 
also conveniently annex the mark of opposition 0, at the left hand, to the symbol of 
any whole number, 7 or n 9 or 0, in order to form a symbol of its opposite number, 
nO, n, or 0; and thus may write 
0 n=n 9, O(n O)=n, 8 O=0; (106.) 
if we still denote by n any positive whole number, and if we call two whole numbérs 
opposites of each other, when they are the determining or multipling numbers of 
two opposite multiple steps. 
14. Two or more multiples such as » x a, v x a, & x a, of the same base a, may 
be compounded as successive steps with each other, and the resulting or compound 
step will manifestly be itself some multiple, such as w x a, of the same common base 
a; the signs n, v, &, denoting here any arbitrary whole numbers, whether positive, or 
contra-positive, or null, and w denoting another whole number, namely the deter- 
mining number of the compound multiple step, which must evidently depend on the 
determining numbers » v € of the component multiple steps, and on those alone, 
according to some general law of dependence. This law may conveniently be de- 
noted, in writing, by the same mark of combination + which has been employed 
already to form the complex symbol of the compound step itself, considered as de- 
pending on the component steps ; that is, we may agree to write 
aod 
