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

168 SINGULAR POINTS OF CURVES.
branches of the curve meet at each of these points, which we
may do as follows.
Tondeter- 240. Let x( =a and y =b be the co-ordinates of any
mine how
many one of these points; then since the substitution of a and b
branches
of acurve for x and y makes
pass
through a dU =o and dyU=o,
which
dU=O, dy
dyU=0. we must proceed to determine - as in (153, &c.), and we shall
dex
dy
arrive at an equation for determining - of a higher order
dx
than the first. If this equation has n real roots, then n
branches of the curve and no more meet at the point (a b).
If any of these roots be equal, then the corresponding branches
have the same tangent at the point (ab). If when ( is a
little greater than a, y has n real values, and when x is a
little less than a only n- 2m real values, (for an odd number of real roots cannot disappear) then 2m of the branches
do not appear on the left side of the ordinate b, and each
two of these branches must therefore either touch the ordinate b as in fig. 67, which will be the case with those
branches for which - becomes co when x =a; or else they
dx
must meet the point (a b) in the form of a cusp as in (fig.
68) or (fig. 69), which will be the case with those branches
for which dy does not become infinite when x = a.
dx
If the 2m real values of y are wanting when x is a
little greater than a, the same is true, only the cusps, &c.
are reversed in position.
dy av root
If the equation for determining d3 have no real root
(a b) is a conjugate point.
Themethod 241. We may often very readily find the nature of a
of ex.pansion in curve near a proposed point (ab) by expanding y- b in powers
oft4enell of -a, by the method given in (145): each different expansion