384 Professor Hamizton on Conjugate Functions, 
In general, we shall denote by the symbol 
Bz, or bs, if b> 0, ~-E0, v20, (325.) 
that positive ratio which is either the only value, or at least one of the values of the 
symbol pu or 6»; and it is important to observe that this positive ratio is not 
changed, when the form of the fractional logarithm S is changed, as if it were a 
fractional multiplier, by the rule (135.), to the form rm or (as it may be more 
concisely written) {’ 3 that is, 
ov 
Bo#=B": (326.) 
a theorem which is easily proved by means of the relation (268.), combined with the 
determinateness (already proved) of that positive ratio which results from powering 
or rooting any proposed positive ratio by any positive or contra-positive whole 
number. 
5th. With respect to the five remaining notations of the group (297.), namely, 
those in which O occurs, we have 
o o 
B@=1; Bem=15 (327.) 
also the symbols 
n 
o 
B°, B 
on 
0 
: (328.) 
are each indeterminate when B=1, and absurd in the contrary case ; and, finally, the 
symbol 
o 
) 
B 
(329.) 
is absurd when eB = 1, but determined and =1, when p=1. 
6th. Proceeding to the group (299.), the symbols 
(0»)7, (@5)%, (8)>, (330.) 
are absurd; the symbols 
(o2)", (OR)om, (331.) 
are determined and each =1, if m be odd, but otherwise, they are absurd; and the 
four remaining symbols 
(Ox)*, a) aa (Cis)e>s (OB) on, (332.) 
