Sead 
and on Algebra as the Science of Pure Time. 419 
anor =( cos S24, sia Om); (164.) 
and therefore the number of distinct values of any fractional power of a couple, is 
equal to the number m of units which remain in the denominator, when the fraction - 
has been reduced to its simplest possible expression, by the rejection of common 
factors. 
15. Thus, the powering of any couple (b,, b:) by any commensurable number « 
may be effected by the formula, 
(5;, 5)? =F (x F—' (5,, by); (165.) 
or by these more specific expressions, 
(b, 6,’ =¥ (wF-'(b,, b,)) 
=i, Bs) (15.0), (166.) 
im which 
(1, 0)'=(cos2wan, sinQwarn): (167.) 
and it is natural to extend the same formule by definition, for reasons of analogy and 
continuity, even to the case when the exponent or number w is tncommensurable, in 
which latter case the variety of values of the power is infinite, though no confusion 
can arise, if each be distinguished from the others by its specific ordinal number, or 
determining integer w. 
And since the spirit of the present theory leads us to extend all operations with 
single numbers to: operations with number-couples, we shall further define (being 
authorised by this analogy to do so) that the powering of any one number-couple 
(b;, bs) hy any other number-couple (2, x») is the calculation of a third number-couple 
(c , ¢2), such that 
(4; C2) = (41, by) (1 ,) =F (@, X,) XPS! (b,, b)); (168.) 
or more specifically of any one of the infinitely many couples corresponding to the 
infinite variety of specific ordinals or determining integers w, according to this for- 
mula, 
(4:5 by) (1 ) =F (in, ) XBT (6,5) 
= (h,, by) (x1; X>) (1, 0) (Xi, Xs), (169.) 
in which the factor (4, 52) (» 2) may be called the principal value of the general 
VOL, XVII. : 4,5 
