in reply to the Observations of Professor Young. 395 
cases.” It is here evident that 1 had not contented myself 
with testing the accuracy of the propositions by the two par- 
ticular examples. On the contrary, my conviction of their 
truth was founded on the solid evidence of mental demonstra- 
tion, and the examples were adduced for the purpose of illus- 
tration without any reference to the proof of the principle 
itself, which, in common with the others, may be established 
without much difficulty. I shall now proceed at once to the 
demonstration of these principles. 
First, then, it is required to be shown that, logically, we 
have no right to reduce a fraction by dividing its numerator 
and denominator by absolute nothing. Let ¢2, $x be two 
functions of a variable 2 which do not vanish when «=a; and 
suppose another variable y to be so connected with x as to 
always fulfill the condition 
(vx—a) Ox—y (2 — al > @ = 0. (1) 
in which 2, 8, are two positive numerical indices and either 
whole or fractional. The value of y deduced from this con- 
dition is 
(v7 —a) Ou 
(x—a)* ox 
and takes the most general form of a vanishing fraction. Sup- 
pose it to be reduced by dividing the numerator and deno- 
minator by (2 — a)’, and it becomes 
te = a) Pe Se PP ed ore 
= ae RON (a — a) Sectoebie (3) 
Let x now be taken equal to a and the expression (2) will 
become y = 2, while (3) will give 
Oo a > ic} 
we 2 if \- = Bo (4) 
in a< fp 
But if we refer back to the original condition (1), it is plain 
that it will be satisfied with 2 = a, independently of the value 
of y, that in this case it imposes no limit whatever on the value 
of y which is therefore completely indeterminate. It follows 
therefore that the result of (2), when x = a, viz. y = 8, must 
have the same indeterminate acceptation ; and that the pro- 
cess of dividing the numerator and denominator of (2) by 
(2 — a)’, (= zero when 2 = a,) which produces (3), and so 
determines a particular value for y, is inadmissible when 2=a, 
and ought not in that case to be performed. And as the ex- 
282 
y= eu Mss. (2) 
