364 Professor Hamitton on Conjugate Functions, 
For if c’ (for example) be the least or hindmost of the ratios a”, 6", c’, 
. so 
that 
<= 
et, OU SboesSw en. (235.) 
and if J (for example) be the greatest or foremost of the ratios a’, b’, c’, ... so that 
EGLO COO LI (236.) 
= denoting what might be more fully written thus, ‘‘ < or =” 
and the other abridged sign = denoting in like manner “> or =”,) then the con- 
ditions (234.) of the Corollary will all be satisfied, if we can satisfy these two condi- 
tions, 
(the abridged sign 
a> OU, we 5 (237.) 
and this, by the Lemma, it is possible to do, if we have the relation 
Coma (238.) 
which relation the enunciation of the Corollary supposes to exist. 
Remark.—lf the ratios a’ b'c... a” bc’... be all actually given, and therefore 
limited in number ; or if, more generally, the least of the ratios a” b” ce’... and the 
greatest of the ratios a’ b’ c’... be actually given and determined, so that we have 
only to choose a ratio a intermediate between two given unequal ratios ; we can then 
make this choice in an indefinite variety of ways, even if it should be farther required 
that a should be a fractional number ”, since we saw, in the 8th article, that be- 
mu 
tween any two distinct moments, such as a’ (B—a)+a and a” (B—A) +A, it is pos- 
sible to insert an indefinite variety of others, such as ~ (s—a) +, wniserial with the 
two moments a and xB, and giving therefore fractions such as ”, intermediate (by 
pe 
the 21st article) between the ratios a’ and a’. But if, instead of actually knowing 
the ratios a’ b' c’... a’ b' c’... themselves, in (234.), we only know a law by which we 
may assign such ratios without end, this law may lead us to conceive new conditions 
of the form (234.), incompatible with some (and perhaps ultimately with all) of these 
selections of fractional ratios ~, although they can never exclude all ratios a what- 
ever, unless they be incompatible with each other, that is, unless they fail to possess 
the relation mentioned in the Corollary. The force of this remark will soon be felt 
more fully. 
Lemma Ill. If b denote any given positive ratio, whether it be or be not the 
