£06 Professor Hamitton on Conjugate Functions, 
such as (0,%, B,) (6,%, 5,)...(6,, b) ; and the number of such products is the 
number of combinations of m things, taken » by n, that is, 
1x2x3x...«xm 
}x2x3~x ...(m—n)x 1x2x3x...0? (71.) 
while these products themselyes become each =(4,, 2,)", when we return to the case of 
equal factors. 
The formula (69.) enables us to determine separately the primary and secondary 
numbers of the power or couple (a, a,)”, by treating the base (a,, a2) as the sum of 
a pure primary couple (@, 0) and a pure secondary (0, a,), and by observing that the 
powering of these latter number-couples may be performed by multiplying the powers 
of the numbers a, a, by the powers of the primary and secondary units, (1, 0) and 
(0, 1); for, whatever whole number 7 may be, 
(a,, 0)'=a;' (1, 07, 
(Fa; 2=ar(Ors) 
We have also the following expressions for the powers of these two units, 
aCe, OO} SGROy 
(0; 1945 SCOR): 
(0, 4) = 1; 0), (73.), 
(1)! S00. = 1), 
(0, 1) “* =(1,0); J 
that is, the powers of the primary unit are all themselves equal to that primary unit ; 
but the first, second, third, and fourth powers of the secondary unit are re- 
spectively 
(72.) 
(0, 1) (—1, 0), (0, —1), (1, 0), 
and the higher powers are formed by merely repeating this period. In like manner 
we find that the equation 
(4, %)"=(b,, be), (74) 
is equivalent to the two following, 
b, =¢, 22 (m—1) G Arnie 4m (n—1) (m—2) (m —3) a,"—"a,'— &¢e 
Sao 1x2x3x4 i i 
(75.) 
m= m (m—1) (m—2) 
b.=m a, (Ca ORa aaa Ga Oa tC. 
For example, the square and cube of a couple, that is, the second and third positive 
powers of it, may be developed thus, 
(a1; a)?={Cam, 0) +0, a) P=(a2— a’, 2a a), * (76.) 
