418 Professor Hamitron on Conjugate Functions, 
or a REAL COUPLE, namely (as we have just now seen) the principal square-root of 
the couple (—1, 0). In the latter theory, therefore, though not in the former, this 
signy —1 may properly be employed ; and we may write, if we choose, for any cou- 
ple (a, az) whatever, 
(a, @)=a,+a,V—}, (158.) 
interpreting the symbols q; and @,, in the expression @, + @, / —1, as denoting the pure 
primary couples (@, 0) (4, 0), according to the law of mixture (61.) of numbers with 
number-couples, and interpreting the symbol /—1, in the same expression, as de- 
noting the secondary unit or pure secondary couple (0, 1), according ,to the formula 
(157.). However, the notation (a, @,) appears to be sufficiently simple. 
14. In like manner, if we write, by analogy to the notation of fractional powers 
of numbers, - 
(a, Cy) = @, b)e, (159.) 
whenever the two couples (%,, 4,) and (¢, ¢:) are both related as integer powers to one 
common base couple (4%, @,) as follows, 
(G, b.) = (a, ds)"; (4a, C2) = (a, a)’, (160.) 
(u and v being any two whole numbers, of which » at least is different from 0,) we 
can easily prove that this fractional power-couple (c¢,, ¢), or this result of powering 
the couple (d,, 5.) by the fractional number a has in general many values, which 
are all expressed by the formula 
(cy )=(by be = F ("e 1b, b,))s 161.) 
and of which any one may be distinguished from the others by the notation 
(by byt = F Bap ' (b,, b,)) (162.) 
We may call the couple thus denoted the w’th value of the fractional power, and in 
particular we may call 
(6, b:) r= r (= F-"(b, b,)) (163.) 
the principal value. The w’th value may be formed from the principal value, by 
multiplying it by the w’th value of the corresponding fractional power of the primary 
unit, that is, by the following couple, 
