ae 
and on Algebra as the Science of Pure Time. 407 
and 
( a, a2)®= {(ai, 0) + (0, a:)}°=( a2 —3 a1 a, 3a, a,— a,°). (77.) 
9. In general, if 
(a, a) (ay a'2)=(a"s, a’2), (78.) 
then, by the theorem or rule of multiplication (54.) 
a") = a, 2 — a, Ay A =, 1 + Ay (79.) 
and therefore 
ay t+ a'y=(a"+ a?) (a+ a7); (80.) 
and in like manner it may be proved that 
mo 
” af (a, a> ) (a, as) (a1, a's) =I (ala taia\s ' (81.) 
then (a+ a"s)=( a+ ar) (a+ a?) (a’24+ a’), 
and so on, for any number m of factors. Hence, in particular, when all these m fac- 
tors are equal, so that the product becomes a power, and the equation (74.) is satisfied, 
the two numbers 0, ), of the power-couple must be connected with the two numbers 
a, a» of the base-couple by the relation 
bP +bP=( a. + a7)". (82.) 
For example, in the cases of the square and cube, this relation holds good under the 
forms 
(a?— a®l+(2a w)’=(a’?+ a)’, (83.) 
and 
(a2—3 a, a?)+ (8a: a— a*)=(a?+ a2). (84.) 
The relation (82.) is true even for powers with contra-positive exponents —7, that is, 
bt +bt=(at+ az)” if (by b)=(a» a@)~"5 (85.) 
for in general 
at 
(41, a2) (a1, a») (@'1,4"2) «. 
UCT, Os) =i, cali eis da) (Cis. eal ; 
ay B2)— (ai? + as?) (a)? + a2?) (a's? + a"2?)... 
then (bh; 3 De ee ara (1? + eo?) (e'1? + co”)... 
