376 Professor Hamitton on Conjugate Functions, 
should confound it with that dangerous and common fallacy, of inferring, in Pure 
Science, the necessary truth of a premiss in an argument, from the known truth of 
the conclusion. 
Resuming the more mathematical part of the research, we may next establish 
this 
Vith Lemma (generalised from IIId): There exists one positive ratio a, and only 
one, which satisfies all the following conditions, 
a> z whenever er <4, 
n" n"\™ (279.) 
a<aa whenever S&S >b; 
6 being any given positive ratio, and m any given positive whole number, while 
mn’ m' n' are also positive but variable whole numbers. The proof of this Lemma 
is so like that of the IIId, that it need not be written here; and it shows that in the 
particular case when the given ratio 4 is the m‘* power of a positive fraction 
n, $ 4 A : ; 
cae then a is that fraction itself. In general, it will soon be shown that under the 
] 
conditions of this Lemma the mt" power of a is 4. 
VIIth Lemma (generalised from IVth). It is always possible to find, or to con- 
ceive as found, two positive whole numbers m, and 7,, which shall satisfy the two 
conditions 
(x) ">e, (“)" <o', if oh >, o>0, Se 
m, } 
m being any given positive whole number ; that is, we can insert between any two 
unequal positive ratios 2’ and 2” an intermediate fractional ratio which is itself the 
m+ power of a fraction. 
For, when m=1, this Lemma reduces itself to the IInd; and when m > 1, the 
m 
theorem (273.) shows that the excess of =) over Ge) may be expressed as 
follows : 
e) (ear + (242)" == xp, * (281.) 
mM, m, 
n e@l-+m n o2+m n e@3+m 1 n a 
dei al ppongs olin ere ea eee 
™, mM, m, m, mM, 
2 
etsy) Gas e ere Fa ee El it (282.) 
m, m,\ m, m, 
in which 
