and on Algebra as the Science of Pure Time. 405 
and 
(1, 0) 
(4, a)'= (4, 4); (4, (as a,) 
se 2 (SF) 
(4, GY = (4, a) (%, &), (a, a2) Gi 2) (dy 2) | 
ec &e. J 
To power the couple (4, a.) by any positive whole number m, is, therefore, to 
multiply, m times successively, the primary unit, or the couple (1, 0), by the proposed 
couple (4, @); and to power (4, a) by any contra-positive whole number —m, 
is to divide (1, 0) by the same couple (4, a), m times successively : but to power by 
O produces always (1, 0.). Hence, generally, for any whole numbers p, v, 
(4, a) u (4, a2) v— (4; ay) uty ae 
CG; 4)" )”=(Can %)*”- : (07.) 
8. In the theory of single numbers, 
(a+b)” _ a” ¥e He b a ans? BP 
1x2x3...xm ~1x2x3...xm 1x2x3x...(m—I) 1 1x2x8x...(m—2) 1x2 a 
a ii um : 
# x FTX aKa xm? (68.) 
1 1x2x8x...(m—1) 
and similarly in the theory of number-couples, 
{(a, ay) ar (A, b,)}” = (4; a,)” (4, ay" Cr Os) 
1X2 XIX 5m 1x2x3x...m UX2X3 xin (n—1) Te 
(4, CDN (4, b,)? + 
+ 1x2x8x...(m—2) 1x2 ws 
ph GT EE rake Gi (69.) 
1x2x3x...(m—1) 1x2x3x...m’ 
m being in both these formula a positive whole number, but @ 4 @, @, 4, 4, being any 
numbers whatever. The latter formula, which includes the former, may easily be 
proved by considering the product of m unequal factor sums, 
(a, 2) F (6, bx"); (a, a>) + Ge; sd) ee (a, a2) as (™, 5, ) > (70.) 
for, in this product, when developed by the rules of multiplication, the power 
(4; a,)"~" is multiplied by the sum of all the products of m factor couples 
