An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

PREFACE.

Vii

of that condition. An example of this is to be found
in a proof often given of the principle of indeterminate coefficients, in which the factor x is divided out
of the equation
Bx + Cxf + Da + &c.... =0,
which of course tacitly assumes the condition that x
is not zero; in this manner is obtained the equation
B+Cx+Dx+ &c. o.. =0;
and then by putting x =0, contrary to the implied
condition, the conclusion B=0 is arrived at. Another
example of this kind of reasoning is given in the first
note, page 6.
The assumption, that f(x + h) can be expanded in
a series of the form A + Bha -- Chp &c.... seems to
me to be a serious defect in the common method of
establishing Taylor's Series, and thereupon the principles
of the Differential Calculus. This assumption is usually
justified by arguing, that if we find definite values for
A, B, C, &c. it shews that the assumption is correct.
Now this argument may be stated thus:
" If the assumption that f(x + h) = A + Bha + Ch+
+&c. be true, then A., B, C, &c., must have definite
values. But we can in general obtain definite values
for A, B, C, &c.... (e.g. by the method of indeterminate coefficients.) Therefore the assumption is true."
This is clearly a falacious argument, for to warrant
the conclusion the first premise should have been this: