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

CHAPTER XVII.
THE GENERAL THEORY OF CONTACT. INTERSECTION OF
CONSECUTIVE CURVES.
Different 242. LET PQ, PQ', PQ", (fig. 30) be three curves having
contact. a common point P, the co-ordinates of which are AM = a,
MP - b; and let the respective equations to these three
curves be
NQ =y =f (), NQ'=y'= ](), ( NQ"=y"= x());
th QQ () - f (x)
then
QQ' q(- ) (I) )
Which, since the three curves meet at P, and therefore
f(a) = (a) = ((a),
assumes the form - when x = a.
0
Hence, by (148), when x approaches a, we have in general,
QQ",+(a) -f'(a)
li. val. of QQ ('(a)-f'(a)
Now suppose that (P'(a) =f'(a), while +'(a) does not
f'(a); then this limiting value is infinite; therefore when
the point N approaches M, QQ" gets continually larger in
comparison with QQ', and may be made as many times larger
than it as we please by sufficiently diminishing MN; and
therefore the curve PQ' must lie infinitely closer to PQ in the
immediate vicinity of the point P than the curve PQ" does.
Hence it appears that if, for the same abscissa, two curves
have the same value, not only of y, but also of -,, they not