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

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only meet, but also lie infinitely closer to each other in the
immediate vicinity of the point of occurse than they would do
dy
if they had not the same value of d at that point.
But suppose that +'(a) -f(a), and ('(a) -f'(a) are
both zero, then by (148) we have in general, when v approaches a,
QQ +2(a) -f (a)
lim. val. of QQ = (a) -pf2(a).
QQ' q)(a) -f (a)
Suppose here that +2(a) =f2(a) while \2(a) does not
=f2(a), then this limiting value is infinite. We may therefore conclude just as before, that if for the same abscissa two
curves have the same values, not only of y and d-, but also
d2
of then they lie infinitely closer to each other in the
immediate vicinity of the point of occurse than they would
d2y
do if they had not the same value of
d xY
And in general, if
q: (a) -f (a), '(a) -f'(a).-..n"-l(a) _ fn-1 (),
(a) - f (a), 4(a) - f (a) '.... n"'(a) -f'-'((a^),
be each zero, we have, when x approaches a,
QQ,,' (a) _ f n (a)
lim. val. of QQ ()
QQI ' V (a) -f (a)
Suppose here that +8(a) f"(a) while "n(a) does not
= f(a); then this limiting value is infinite. We may therefore conclude that if for the same abscissa two curves have the
dy d"y dxy
same valtzes of y d —, d... d —9 then they lie infinitely
closer to each other in the immediate vicinity of the point of
occztrse than they would do if they had not the same value of
dny
d x"