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

106 METHOD OF FINDING - FROM
dx
Thus it appears that where the first differentiation of
dy
an equation fails to give us -d for a particular value of x,
in consequence of all the terms becoming zero, we must differentiate successively until we come to an equation, all the
terms of which do not vanish when we give x that particular
value.
These tt54.. If we perform these differentiations on the supdifferentiations may position that p is a constant, we shall arrive at a correct
be per- result: for it is easy to see from the above examples that
the suppo- all the terms obtained by differentiating p once or oftener
sition that
p is con- vanish when we give x the particular value for which we
stant. wish to find p, and therefore do not affect our final equation,
which gives us the value of p. Hence we may always differentiate on the supposition that p is constant, and this will
somewhat simplify the process.
A some- tt155. The values of p are sometimes found by the
what dfferent method method of finding the limiting value of a vanishing fraction
of findnge given in (148); thus in the first example we have
for vanish-+
ing frac- 2 c3+ oa x - 2 a2y 00
tions. P 2 = when =
2as 4 'o a y O0
therefore by (148),
6V2 - 3a2 - a2cp 3-2p
6-' a - =a= - when = 0;
a2+ a"p 2 -2- p.'. 2p +p2= $ - 2p, p + 4p -3 = 0,
which is the result we arrived at before.
Another tJ- 156. The following is an example where p assumes
example. 0
the form - for other values of v and y besides zero.
0
To find p when v = a from the following equation, viz.
ay2_ 2 a2y- V3 + 3aX2 = 0..... (I).
Differentiating, we have
(2ay - 2a) p - 6i2 + 6ax = 0.