366 Professor Haminton on Conjugate Functions, 
posed ratio 4. In like manner it may be proved that in the other case, when @ is the 
square of a positive fractional or positive whole number a; one positive ratio @ and 
only one, namely the number = itself, will satisfy all the conditions (239.) ; in both 
cases, therefore, the Lemma is true: and the consideration of the latter case shows, 
that, under the conditions (239.), 
eh ae eee SO: (246.) 
m mm me 
In no case do the conditions (239.) exclude all ratios a whatever ; but except in the 
case (246.) they exclude all fractional ratios : for it will soon be shown that the one 
ratio a which they do not exclude has its square always =b, and must, therefore, be 
an incommensurable number when # is not the square of any integer or fraction. 
(Compare the Remark annexed to the Corollary of the IInd Lemma.) 
Lemma IV. If b' and 6" be any two unequal positive ratios, it is always possible 
to insert between them an intermediate fractional ratio which shall be itself the square 
. ere = . 
of another fractional ratio a that is, we can always find, or conceive found, two 
positive whole numbers m and » which shall satisfy the two conditions, 
nan 
ey ag a moe if 6 > b, 6>0. (247.) 
For, by the theorem of multiplication (200.), the square of the fraction : _- may 
be expressed as follows, 
a ! fo) ' 7 ok 
Ua Cee Pe ee” Pade (248.) 
m m mm mem mm 
Sout? As Sc aa | ON epee 
that is, its excess over the square of the fraction — 1s —_ + , which is less than 
9 m mm mm 
2 1+ : ; S40 ; 
Sie og a , and constantly increases with the positive whole number 7’ when the 
m m 
positive whole number m remains unaltered ; so that the 1+ squares of fractions 
with the common denominator m, in the following series, 
1 1 2 2 3} 3 n' n! lin 1lxn 
== x= Dee Sl Set a z (249.) 
m m m m m me mm m m Ww 
5 5 . o 5 I 2 1 + n d I f 
increase by increasing differences which are each less than a} = therefore 
Lieve ; : ae 
than i? if we choose m and m’ so as to satisfy the conditions 
m=2Zik, 1+n'=im, (250.) 
