396 Mr. Woolhouse on the Theory of Vanishing Fractions, 
pression (2) comprehends every possible case of vanishin 
fractions, the reasoning is general, and fully establishes the 
position occupied in the first extract. 4 
It is here evident that the same objection will apply to the 
division by zero of an equation involving two variables, or that 
the equation resulting from a division by a common factor is 
inadmissible when that factor absolutely vanishes. Thus the 
equation (1), when divided by (2 — a)F, gives 
(x —a)* "02—you =0, 
which would, for « = a, give to y the particular value in (3), 
a circumstance quite inconsistent with the nature of the con- 
dition involved in the antecedent equation (1), which, in the 
case x = a, places no restriction on the value of y. It is also 
obvious that the multiplication of an equation, or of the nu- 
merator and denominator of a fraction, by zero, is equally ob- 
jectionable, as regards propriety of reasoning, since, by that 
process, we might pass from conditions that determine par-. 
ticular values to others of a totally indeterminate character. 
Before quitting this. point it will be well to draw a general 
and necessary inference that may, in conjunction with the 
fourth extract, contribute in some degree towards the eluci- 
dation of the present inquiry. It is this: —That when a quan- 
tity, which we know from other considerations ought to have 
a determinate value, comes out in a vanishing fraction, or, 
vice versa, when a quantity, which we know to be indetermi- 
nate, comes out in a determinate form, we may be assured: 
that at least one of the steps, in the process of solution, fails 
in the manner here explained. : 
The proof of the principles contained in the other extracts 
immediately follows from the preceding demonstration. Sup- 
pose the equation (2) to express the result of an analytical 
investigation in which the reasoning throughout is admissible 
when x = a, so that no multiplication or division by a power 
of x —a occurs in the process. We proceed to show that 
the resulting vanishing fraction (2), when x =a, must be in- 
determinate in value. The equation (1), which is antecedent 
to, and corresponds in signification with, the equation (2), is 
satisfied with 2 = a, without any reference to the value of y, be- 
cause that equation is divisible by a positive power of « —a. 
Since, therefore, in the investigation, no multiplications or divi- 
sions have been made by 2 —a or any power of it, it is conclu- 
sive that the series of equations, which precede the equation (1) 
in the course of reduction, must likewise be divisible by the same 
power of z — a, and therefore be satisfied with x = a, inde- 
pendently of the value of y. The primitive equation from 
