404 Professor Hamitton on Conjugate Functions, 
in which 
2(h, bs) (a; a.) =(2, 0) (di, b:) (a; a.) =(b,, bz) (iy G2) + (h,, b.) (Gis Ga) 5 (60.) 
for, in general, we may mia the signs of numbers with those of number-couples, if we 
consider every single number a as equivalent to a pure primary number-couple, 
=O) (61.) 
When the pure primary couple (1, 0) is thus considered as equivalent to the number 
1, it may be called, for shortness, the primary unit ; and the pure secondary couple 
(0, 1) may be called in like manner the secondary unit. 
We may also agree to write, by analogy to notations already explained, 
(0, 0) +(a; a2) =+ (Q; a), 
(62.) 
(O, 0) —(a, a) = (a, a’) 3 
and then + (a, @,) will be another symbol for the number-couple (a, a) itself, and 
—(a, a) will be asymbol for the opposite number-couple (—a, —a,). The reciprocal 
of a number-couple (a, “) is this othe#number-couple, 
1 =o -( a =a )=o=@) (63.) 
— nl 2 Pe 2 2 ag "| nae 
(Qj, Gs) (%, 4%) ay + 4% ay +4, ay + az 
It need scarcely be mentioned that the insertion of the sign of coincidence = 
between any two number-couples implies that those two couples coincide, number with 
number, primary with primary, and secondary with secondary ; so that an equation 
between number-couples is equivalent to a couple of equations between numbers. 
On the Powering of a Number-couple by Single WV hole Number. 
7. Any number-couple (a, a.) may be used as a base to generate a series of 
powers, with integer exponents, or logarithms, namely, the series 
o'er, (a, ay)~*, (a, Q)—"; (a, a), (a, as)', (a, as)", eee (64.) 
in which the first positive power (a, a)' is the base itself, and all the others are 
formed from it by repeated multiplication or division by that base, according as they 
follow or precede it in the series ; thus, 
(a, a2)’ =(1, 0), (65.) 
