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

APPENI)1X. 217
an a/3 ((y d - xd y)2
and - or -
2 2dxdy
is to be a maximum.
Therefore, considering dx as constant, we have
(ydx.- xdy )2Y
or y =0,,
dy
- 2 (ydo - zdy) ovdfydy - (yfdsc - xdy)2d~y
or _ 0,
dy
ydx - xdy
which gives 2x - = -a;
a
2
this equation shews that the portion of the tangent intercepted
between the co-ordinate axes is bisected at the point of contact
when it cuts off a maximum or minimum area.
(3) In the ellipse, if we assume x = a cos t, then we have
y = b sin t, in virtue of the equation
y2 w_
b-2 = 1 a2
Hence we may always suppose in the ellipse, that
x = a cos t, and y = b sin t.
If we describe a circle on the major-axis as diameter, and if
we produce the ordinate MP to meet this circle in Q, then
it is evident that z QCA4 = t, for
CM x
cos QCA = - = cos t,
MQ a
This transformation is often useful.
Differentiating, we have
dx = - asin tdt, dy = b cos t dt;