382 Professor Hamitton on Conjugate Functions, 
are absurd, or denote no ratios whatever. In like manner the symbol 
1 v 
0®™, and more generally Oo», (310.) 
is absurd, or denotes no ratio, because no ratio a can satisfy the equation 
a°™=0. (311.) 
We have thus discussed all the nine cases (298.), of powers in which the base is 0, 
and haye found them all to be impossible, except the two first, in which the exponents 
are ~, and ~, and in which the resulting powers are respectively 0 and 1. We. 
have also obtained sufficient elements for discussing all the other cases (297.) and 
(299.), with their sub-divisions (300.), as follows. 
lst. B™ is determined and positive, unless m is even, and x odd; in which case 
: e1+2k 3 : 4 Z . 
it becomes of the form B —g:—, and is ambiguous, being capable of denoting either 
of two opposite ratios, a positive or a contra-positive. To distinguish these among 
themselves, we may denote the positive one by the symbol 
eo1+42k 
Bist cs (312.) 
and the contra-positive one by the symbol 
0 142k 
OB ar 3 (313.) 
for example, the two values of the square-root /B or B*, may be denoted for dis- 
tinction by the two separate symbols 
i oy 
B?=/8B, OB®=0,//B. (314.) 
° 
The other three cases of the notation B”, namely, the symbols 
olpzk _2k 2k 
polt2i ,pelfai py Bi (315.) 
> 
denote determined positive ratios. 
2d. The three cases 
1+0 (2k) 0 (2k) © (2 k) 
p OM? pelei , 8t) (316.) 
on 
of the notation B ™, are symbols of determined positive ratios; but the case 
