398 Professor HamintTon on Conjugate Functions, 
when the two step-couples (b,, b:) (c:, «) are related as multiples to one common 
step-couple (a,, a.) as follows : 
(bi, Ne) = tees (Gyn ai); (e1, Cn p= yd (Canstan)5 (17.) 
» and » being any two proposed whole numbers. And if we suppose the fractional 
multiplier ~ in (16.) to tend to any incommensurable limit a, we may denote by 
@ x (bj, b,) the corresponding limit of the fractional product, and may consider this 
latter limit as the product obtained by multiplying the step-couple (b,, b,) by the in- 
commensurable muitiplier or number a; so that we may write, 
(ep eo) =a x (b1, bo) = a(b, bo), 
‘i us) 
if (cj, cs) = L (“(b, be)) anda=L~, 
Bh B 
using L as the mark of a limit, as in the notation of the Preliminary Essay. It follows 
from these conceptions of the multiplication of a step-couple by a number, that gene- 
rally 
aX (aieaa) = (aa, a as)s (19.) 
whatever steps may be denoted by a,, a, and whatever number (commensurable or 
imcommensurable, and positive or contra-positive or null) may be denoted by a. 
Hence also we may write 
ue ts ? aS (20.) 
and may consider the number a as expressing the ratio of the step-couple (a 9), @ s,) 
to the step-coup!e (a1, 22). 
On the Multiplication of a Step-Couple by a Number-Couple ; and on the Ratio of — 
one Step-Couple to another. 
4. The formula (20.) enables us, in an infinite variety of cases, to assign a single 
number a as the ratio of one proposed step-couple (b,, b») to another (a), a2); 
namely, in all those cases in which the primary and secondary steps of the one couple 
are proportional to those of the other : but it fails to assign such a ratio, in all those 
