356 Professor HamiLton on Conjugate Functions, 
a and b’ being then the extremes, and » the mean, or the mean proportional between 
a and b’, in this continued analogy ; in which w is also the third proportional to a 
and pv, and a is at the same time the third proportional to v and b, because the 
analogy may be inverted thus, 
a b 
eee (202.) 
When the condition (201.) is satisfied, we may express b! as follows, 
b 
Ss Seg (203.) 
a 
that is, if we denote by a the ratio of b to a, we shall have the relations 
Di=texga', De=—teeee bi == e (204.) 
and reciprocally when these relations exist, we can conclude the existence of the con- 
tinued analogy (201.). It is clear that whatever effective steps may be denoted by 
a and b, we can always determine, (or conceive determined,) in this manner, one 
third proportional b’ and only one; that is, we can complete the continued analogy 
(201.) in one, but in only one way, when an extreme a and the mean b are given: 
and it is important to observe that whether the ratio a of the given mean b to the 
given extreme a be positive or contra-positive, that is, whether the two given steps 
a and b be co-directional or contra-directional steps, the product a x a will necessarily 
be a positive ratio, and therefore the deduced extreme step v’ will necessarily be 
co-directional with the given extreme step a. In fact, without recurring to the 
theorem of the 21st article respecting the cases in which a product of contra-positive — 
factors is positive, it is plain that the continued analogy requires, by its conception, 
that the step b’ should be co-directional to b, if b» be co-directional to a, and that b’ 
should be contra-directional to » if b be contra-directional to a; so that in every 
possible case the extremes themselves are co-directional, as both agreeing with the 
mean or both differing from the mean in direction. Jt ts, therefore, impossible to 
insert a mean proportional between two contra-directional steps ; but for the same — 
reason we may insert either of two opposite steps as a mean proportional between 
two given co-directional steps ; namely, either a step which agrees with each, or a 
step which differs from each in direction, while the common magnitude of these two 
opposite steps is exactly intermediate in the way of ratio between the magnitudes of 
the two given extremes. (We here assume, as it seems reasonable to do, the con- 
ception of the general existence of such an exactly intermediate magnitude, although 
the nature and necessity of this conception will soon be more fully considered.) For , 
