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

228

APPENDIX.

(U.)
(1) To determine the position and nature of the multiple points of the curve,
U = y4 + 4 - 2a2y2-_ 2a2x + a4 = 0... (),
d U = 4 -3 - 4a2Xw = o. @@4@@@(2),
dyU= 4y3 - 4a2y = O
which equations are satisfied by any two of the values
o = or - a, and y = or: a.
Now if. = o, y = - a in virtue of (1),
and if m = L-a, y= o or aV/2.a.
Hence the only values of x and y which satisfy (1) and
(2) are
= 0- 0 -o;O () - )x =) - a)
(~) (j), (7),,, )'
y= aa y = -0 a y = y=
Now d2 U = 12<2 - 4a2,

ddyd U = o,
dy2 U 12y2
Hence, using the notation i
A = o, B = o,
C = - 4a2 if x - 0,
D = 0,
E = - 4a2 if y = 0,
Hence for the values (a)

- 4 a2.
in (161), we have
or 8a2 if x = - -a,

or 8a2 if x = - a.
and (/), we have

dy 1
-4aa2 + 8a2u2 =, and. UQ or d ==
dv '2/'