4.00 Professor HamiLtron on Conjugate Functions, 
whatever numbers may be denoted by a, Oy b, b,, and whatever steps by a, a, by by 
With these conditions we have 
(a, ds) (4, a)=(a, O) (4, ay +(0, az) (4, ay), (29.) 
(0, ) (4, a.) = (0, a) (4, 0) + (0, a2) (0, a), (30.) 
and, therefore, by (22.) and (24.), and by the formula for sums, 
(Mm, ay) (41, a)=(a4 Ao, Gy 44) +(0, ay a) +(0, a») (0, ay) 
=@ 4, QA %+Q, a,)+(0, a2) (0, a), (31.) 
in which the product (0, a.) (0, @,) remains still undetermined. It must, however, 
by the spirit of the present theory, be supposed to be some step-couple, 
(0, as) (0, a )y=(¢, oy) 5 (32.) 
and these two steps ¢, ¢. must each vary proportionally to the product a, a,, since 
otherwise it could be proved that the foregoing conditions, (27.) and (28. 4 would not 
be satisfied ; we are, therefore, to suppose 
(= ¥J1 Ay % “2= Yo Az Ag, (33.) 
that is, 
(0, @2) (0, #2)=(yr Ge M5 Ya Ga %)s (34.) 
yi ys being some two constant numbers, independent of a, and a, but otherwise 
capable of being chosen at pleasure. Thus, the general formula for the product of a 
step-couple («,, @,) multiplied by a number-couple (a, a), is, by (31.) (34.) and by 
the theorem for sums, 
(a, Q) (41, 2)= (a, a, a 4+, 4) + (yy Gs 9% Y2 A a,) 
= (M1 + Yr Ay My Gy A + Ay + Y2 Ge %): (35.) 
and accordingly this formula satisfies the conditions (27.) and (28.), and includes the 
relations (22.) and (24.), whatever arbitrary numbers we choose for y,, and y, ; pro- 
vided that after once choosing these numbers, which we may call the constants of mul- 
tiplication, we retain thenceforth unaltered, and treat them as independent of both 
the multiplier and the multiplicand. It is clear, however, that the simplicity and 
elegance of our future operations and results must mainly depend on our making a 
simple and suitable choice of these two constants of multiplication ; and that in making 
