388 Professor HamItton on Conjugate Functions, 
because, by the theorem of multiplication (273.), or (281.), 
m 
] ] | ee: 1. S9l+m ; 
Ol + Cts) =p {l+atp+aep t+. F04}) - (347.) 
If then c c¢’ be any two proposed unequal positive ratios, of which we may suppose 
that c’ is the greater, 
rae ono 5 (348.) 
we may choose two positive whole numbers 7, k, so large that 
bi>c,2<Octe, (34 9.) 
and two other positive whole numbers J, m, large enough to satisfy the conditions 
(341.) (343.) ; and then we shall be sure that some one at least, such as 4”, of the 
fractional powers of 4 comprised in the series (339.) will fall between the two proposed 
unequal ratios ¢ c’, so that 
ce<bm,c >in, (350.) 
If then the one ratio c satisfy all the conditions (336.), the incommensurable number 
a must be <},, and therefore, by the 2nd relation (350.), the other ratio c’ cannot also 
satisfy all the conditions of the same form, since it is > 8 m, although a<;. In like 
manner, if the greater ratio c’ satisfy all the conditions of the form (336.) the lesser ratio c 
cannot also satisfy them all, because in this case the incommensurable number a will be > ~, 
No two unequal positive ratios, therefore, can satisfy all those conditions : they are there- 
fore satisfied by one positive ratio and only one, and the symbol bs (interpreted by. 
them) denotes a determined positive ratio when b > 1. For a similar reason the same 
symbol 6", interpreted by the conditions (337.), denotes a determined positive ratio 
when 3 < 1, 6 > 0; and for the remaining case of a positive base, 5 = 1, the symbol 
b* denotes still, by (338.) a determined positive ratio, namely, the ratio 1. The ex- 
ponent or logarithm a has, in these late investigations, been supposed to be incom- 
mensurable ; when that exponent a is commensurable, the base é being still positive, 
we saw that the symbol % can be interpreted more easily, as a power of a root, and 
that it always denotes a determined positive ratio. 
Reciprocally, in the equation 
B= lio (351.) 
° 
if the power c he any determined positive ratio, and if the exponent a be any deter- 
