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

EQUATIONS WHEN TAYLOR S SERIES FAILS.

95

or of r, &c., which satisfies the above conditions, in which generally
obtained in
case we find more than one perfect developement for y. this manner.
An example will be necessary to make the nature of this Example.
process of developement clear.
Let y3 - 3axy + X3 = o,
then y =0 when x = 0; assume.'. y = uX,
and.'. substituting
u3xIm - 3Cauv+l + x3- 0... (1).
Now here the powers of x cannot be all different, since the
coefficients of them are, either constant quantities not equal to
zero, or quantities which we suppose not to approach zero as
their limiting value when x approaches zero: for it is clear
that if all the powers of x be different, and if the equation
be divided by the lowest power, and x then put equal to
zero, the coefficient of the lowest power must necessarily
become zero, contrary to hypothesis.
Since then the powers of x cannot be all different some two
of them must be the same; either the first and second, or the
first and third, or the second and third.
(1) Suppose that the first and third are the same; then
3m =m + 1, and -. m; = then (1) becomes, dividing
out xa,
u3 - 3au + a positive power of x = 0,
which gives u=0 or V/3a when =0o: we must of course
reject u=0, since we want only those values of u which are
neither o nor o.
(2) Suppose that 3m 3, and.'. m = 1, then we have
u + 1 + a negative power of x = 0,
which.'. gives u = co when x= o; this value of m must therefore be rejected.