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

APPENDIX.

229

and for the values (y) and (S), we have
dy
8a2 - 4a2u = 0, and.*. u, or d" = /2.
dx
Hence if we take AB = AE = AC = AD = a (fig. 40), there
is a double point at B, at C, at D, and at E, as is represented in the figure.
This is an example which may be very easily solved by
the method in (156).
For in this case
P =_. -;?3 - a2y
3 X2 - a3x22 - a2
' p (3y2_ a2)p.*. if =, and y = a,
p2,
and if - =: a, and y o,
2 = 2.
In general the method given in (159) ought to be used,
dy
only when we wish to find dy for the values x = o, y = o,
da2
or when more than two differentiations are necessary to find
dy
dy in which cases it is simpler than the common method.
(2) Let U = ay - bX2y + x4 = 0... (1),
then dU = - 2bxy + 43 =... (2),
dyU Says - b,2 = o........ (3).
(2) gives x=o or y b '
if x = 0 y= 0 in virtue of (3) and these values satisfy (1),
2 ) becomes
if -= V (3) becomes