416 Professor Haminron on Conjugate Functions, 
On the Powering of any Number- Couple by any Single Number or Number- Couple. 
13. Resuming now the problem of powering a number-couple by a number, we 
may employ this property of the exponential function F, 
(F (a, a:))"=F(u di, 1 a), (142.) 
» being any whole number whether positive or contrapositive or null; which easily 
follows from (125.), and gives this expression for the »’th power, or power-couple, of 
any effective number-couple, 
(8, 6,)*=F(n FCG, , b,)). (143.) 
Reciprocally if (a, a2) be an mth root, or root-couple, of a proposed couple (0, , bs), 
so that the equation (74.) is satisfied, then 
1 1 
(a, a) =(, bo) =F (— F(h,, b)). (144.) 
This last expression admits of many values, when the positive whole number m is > 1, 
on account of the indeterminateness of the inverse or logarithmic function r~'; and 
to specify any one of these values of the root-couple, corresponding to any one value 
¥-! of that inverse function, which value of the root we may call the wth value of that 
root, we may employ the notation 
1 
6, m= F (EG, %); (145.) 
o 
we may also call the particular value 
Bs 1 
(, &)m = F(— FG, 4)), (146.) 
the principal value of the root-couple, or the principal m’th root of the couple @,, 4). 
In this notation, 
2 
wT 3 
m 
Oh y (0, (147.) 
Gitge @,,8)% Gils Oyns (148.) 
so that generally, the w*th value of the mth root of any number-couple ts equal to 
the principal value of that root multiplied by the wth value of the mth root of the 
