354 Professor Hamitton on Conjugate Functions, 
tiply all by any common multiplier a, whether positive or contra-positive, and may 
establish the theorem, 
axe=(axb)+(a xa), if c=b+a; (191.) 
which gives, by the definitions (169.) (170.) for the sum and product of two ratios, 
this other important relation, 
ax (b+b)=(ax 8) +(ax B), (192.) 
if b, b', and 6'+4, denote any three positive or contra-positive numbers, connected 
with each other by the definition (169.), or by the following condition, 
(6 +b) xa=(b'x a)+(b xa), (193.) 
in which a denotes any arbitrary effective step. The definitions of the sum and 
product of two ratios, or algebraic numbers, give still more simply the theorem, 
(b +b) x a=(0 x a)+(b x a). (194:.) 
The definition (169.) of a sum of two ratios, when combined with the theorem 
(75.) respecting the arbitrary order of composition of two successive steps, gives the * 
following similar theorem respecting the addition of two ratios, 
b+a=a+b. (195.) 
And if the definition (170.) of a product of two ratios or multipliers be combined 
with the theorem (186.) of alternation of an analogy between two pairs of steps, in 
the same way as the definition of a compound step was combined in the 12th article 
with the theorem of alternation of an analogy between two pairs of moments, it 
shows that as any two steps a, b, may be applied to any moment, or compounded 
with each other, either in one or in the opposite order, (b +a=a+b,) so any two 
ratios a and 6 may be applied as multipliers to any step, or combined as factors of a 
product with each other, in an equally arbitrary order; that is, we have the relation, 
DO GGs—NCEXE0s (196.) 
It is easy to infer, from the thorems (195.) (196.), that the opposite of a sum of two 
ratios is the sum of the opposites of those ratios, and that the reciprocal of the pro- 
duct of two ratios is the product of their two reciprocals ; that is, 
6 (b+a)=906+0a, (197.) 
and 
u (bxa)=ubxua. (198.) 
