374: Professor Hamitton on Conjugate Functions, 
of the 26th and 27th articles, that for any proposed positive ratio 6 whatever, and 
for any positive whole number m, it is possible to determine, or conceive determined, 
one positive ratio a, and only one, which shall have its m’th power =2; and for this 
purpose to show that the power a” increases constantly and continuously from zero 
with a, so as to pass successively, but only once, through every state of positive 
ratio 5. On examining the proof already given of this property, in the particular 
case of the power a’, we see that in order to extend that proof to the more general 
case of the power a”, we have only to generalise, as follows, the Ist, IJId, and [Vth 
Lemmas, and the Corollary of the Ist, with the Theorem resulting from all four, 
retaining the IInd Lemma. 
Vth Lemma: (generalised from Ist.) 
> > 
iW y=; and 2>0; y>0; then y™ = 2. (272.) 
> < 
When m=1, this Lemma is evident, because the first powers y’ and x* coincide with 
the ratios y and 2. When m > 1, the Lemma may be easily deduced from the con- 
ceptions of ratios, and of powers with positive integer exponents; it may also be 
proved by observing that the difference 0 x” +y”, between the powers x” and y”, 
in which the symbol © a” denotes the same thing as if we had written more fully 
© (#”™), and which may be obtained in one way by the subtraction of 2” from y”, 
may also be obtained in another way by multiplication from the difference 6 a +y 
as follows : 
Oa™+y"™=(Oaty) x(Foltry? +a my tt. Hay ort 1a Yel tm), (B78:) 
and is, therefore, positive, or contra-positive, or null, according as the difference 
© «+y of the positive ratios 2 and y themselves is positive, or contra-positive, or 
null, because the other factor of the product (273.) is positive. For example, 
Oawi+yi=(Ort+y) x(e*+ayty’); (274.) 
and, therefore, when a and y and consequently #*+2y+y?* are positive, the dif- 
ference © «*+y* and the difference © 2+y are positive, or contra-positive, or null 
together. 
As a Corollary of this Lemma, we see that, conversely, 
> > 
if y"=a”", and 4 >0, y> 0, then y =. (275.) 
< 
