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.

III1

dy assume the
then putting i = a + i, and y= b ~ y, in the given equation div, forma
dyl not zero.
will be the same thing as d, and = 0Y, = 0 will be the
dx
dy, assume the form 0; w
values of iv, and y, which make, assume the form -; we
dxv 0
may therefore proceed as before. If the given equation be
U = 0,
and if we denote by A, B, C, D, E... &c. the values of d, U,
dyU, dx2U, 2dxdyU, dy U, &c. when iv is put = a and
y = b; then, by 146, the result of substituting a + xi for iv
and b + y, for b will be
Axiv, By,~ C iv2 Divyl + Ey,2 2 &C.... 0;
if therefore A and B be not both zero, we have
A + Buo = o,
and if A and B be both zero we have,
C+ Duo+~Euo2 =0,
and so on............
and we may find Ue therefore just as before.
Let the given equation be (see 156) Example.
U = ay' - 2 a'y - 2X3 + 38aiv = 0,
dy
from which we wish to find - when i = a, and.. y = b,
div
then d. U =-6X2 6aio = 0when x = a,
dU= 2ay - 2a' = 0 when y = a,
d-1 2U= -_12X + 6a 6 a when x =a,
drVdyU = o,
dy2 U = 2a.
Hence A = o, B- o. C=- 6a, D = o, E = 2a, and
therefore we have
3 + ue2= 0, and.-. ue = 4/3 as before.