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

94

DEVELOPEMENT OF FUNCTIONS GIVEN BY

Hence putting x = 0, we find
from (1) y3= 1;.. y= 1, (at least 1 is one of the
3 values of y);.from (2) Sp +1 =; - -;
6 2 4.*. from (3) 3q + -+-=;.'. q=9 3 9
4 2 38.. from (4) 3r +9.-+-=;.. - - =
9 9 27
&c. &c.
Hence by Taylor's series, we have
1 x 4,2 38 x3
y =1 - --.. &c.
3F1 9 2 27 r3
low we tt- 145. It often happens however that we thus get inceedwhen finite values for some of the quantities p, q, r, &c., and of
Teriefilss course this method fails; we must then proceed as follows.
Let a be a value (or one of the values) of y when
x = 0, then substitute a + uxa for y in the given equation
f(y) = 0, and, if possible, give m such a value, that the
limiting value of u when x approaches o shall not be 0 or co,
but some quantity, b suppose. Then assume u = b + va",
substitute this value of u in the equation and so determine n,
if possible, that the limiting value of v when x approaches
zero shall not be o or co, but some quantity, c suppose. Then
assume v = c + wxP; it is clear that in this manner we obtain
a perfect developement for y, viz.,
y = a + bam + cxrm+ + R, putting R = wx"+P,
which we may carry as far as we please.
Several dif- It generally happens that we obtain more than one value
erent seriesf y when = 0, and more than one value of m, or of n,