386 Professor Hamitton on Conjugate Functions, 
termined integer numbers, (setting aside the impossible or indeterminate case of root- 
ing by the number 0,) we have the relation 
(i ileal Jaleo (334.) 
which is analogous to (267.); and the relation 
cf — exe ifc = DF, (335.) 
analogous to (268.): a and f denoting here any two commensurable numbers. And 
it is easy to see that while the fractional exponent or logarithm a increases, advancing 
successively through all fractional states in the progression from contra-positive to 
positive, the positive ratio b “ increases constantly if 5 >1, or else decreases constantly 
if b<1, ()>0,) or remains constantly = 1 if 6= 1. But to show that this increase 
or decrease of the power with the exponent is continuous as well as constant, we must 
establish principles for the interpretation of the symbol be when a is not a fraction. 
When a is incommensurable, but 6 still positive, it may be proved that we shall still 
have these last relations (334.) and (335.), if we interpret the symbol be to denote 
that determined positive ratio ¢ which satisfies the following conditions : 
whenever a >=, 
m whenever a < x, f (836.) 
if b>1; J 
c = bs < b® whenever a >=, 
(337.) 
c = b* > bw’ whenever a <=, 
if b< 1, b>0; 
or finally this equation, 
c= h=1,if 6-1. (338.) 
The reader will soon perceive the reasonableness of these interpretations ; but he may 
desire to see it proved that the conditions (336.) or (337.) can always be satisfied by 
one positive ratio c, and only one, whatever determined ratio may be denoted by a, 
and whatever positive ratio (different from 1) by 4. ‘That at least one such positive 
