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

AN EQUATION WHICH BECOMES ILLUSORY. 105
But differentiating the equation (2) we have In such a
case we
d2y must differ6xv~ + 3a- 2 ap - (2a2+ a2p) p- (2 a'x 4. a2y) q = 0 (q = entiate
drv' again in
order to find
and putting =0, and.-. y 0, here we find dy
d
3a - 2a2 - (2a'~ a2p) p =0,
or,? L p - 3 = 0;
p=-2 \V4 + 3,
and thus by differentiating twiCe we obtain p, and we find that
it has two values - 2 T "/_7 and -2- 7.
As another example, suppose that we wish to obtain p Another
when v = 0 from the equation example.
X4 + a y3 - 2 av y' - 0cax2y = 0.........(1),
differentiating we find
(43v- 2ay-6axy) + (3ay2 - 4ay - 3a2) po. (2),
here put x 0, and.-. y = 0 in virtue of (I), and we find
0
0
But differentiating again we have
- y -4(2y + 3r) p +~2 (3y- 2 x)p%- (3y2- 4(xy - 3x1) q = 0,
0
here put x =o0, and.-. y 0, and we find again p = L.
0
Differentiating again, therefore, we have
0 = - 18p - 12p9-pl 6p3 terms multiplied by Lv or y,
here putting X = 0 and y = 0, we find
P3 -2p2 - 3p = 0,
which gives p = 0 or p = 1 aV/1+ 3 = 3 or - 1.
And thus by differentiating three times we obtain p, and
we find that it has three values 0, 3 and -1i.