378 Professor HamiLTon on Conjugate Functions, 
ever large may be the given greater positive ratio 4’, and however little may be its 
given excess over the lesser positive ratio 2’. 
Hence, finally, this Theorem : 
n' n'! 
Ifa>—,anda< —,, ] 
Y m! 
n' m n” ™m 28 
whenever (|) < 4, (-) aE b (289.) 
1 
then a" = 0, a="/b =5™; 
5 denoting any given positive ratio, and m any given positive whole number, 
while m' 2’ m" n" are any arbitrary positive whole numbers which satisfy these 
conditions, and a is another positive ratio which the VIth Lemma shows to be 
determined. 
For if a” could be >4, we could, by the VIIth Lemma, insert between them a 
™ 
positive fraction of the form (=) > such that 
Ci) o> (ZB) "<a", (290) 
and then by the Corollary of the Vth Lemma, and by the conditions (289.), we 
should deduce the two incompatible relations 
a SG axm, (291.) 
which would be absurd. A similar absurdity would follow from supposing that a” 
could be less than 4; a” must therefore be =4, that is, the Theorem is true. It 
has, indeed, been all along assumed as evident that every determined positive ratio a 
has a determined positive mt power a”, when m is a positive whole number ; which 
is included in this more general but also évident principle, that any m determined 
positive ratios or numbers have a determined positive product. 
Every positive ratio 6 has therefore one, and only one, positive ratio a for its 
m‘ root, which is commensurable or incommensurable, according as 4 can or 
cannot be put under the form (=) ; but which, when incommensurable, may be 
theoretically conceived as the accurate limit of a variable fraction, 
aaah ye if (By CRED TES (292.) 
m 
SS es a 
