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

BY MEANS OF DIFFERENTIATION.

183

d2u = 0,
dnu = 0.
m + n.m -{*n - 1... m + 1
Now there are m + (= N* suppose)
1............... m
different combinations of the m + n constants a, a2... &c.;
therefore there are N different ways in which we may form an
dy d"y
equation containing a, y,...- and m of the constants;
and consequently we may in general obtain N different equations involving mV y d and m of the constants from
tin in dxn
the given equation.
In the example just given, if we eliminate a between (1) Example.
and (2), we find
a = P from (2);
2 - b
and therefore, by (I),
poa (r - b)
Y- 2-b 0.....* (5).
Again, if we eliminate b, we obtain
y -pv + a2 = 0......... (6),
and thus we deduce from (1) two different equations (5) and
(6), each involving 'x, y, d-' and one of the constants.
256. We may eliminate the constants a1, a, a3... an Thiselimifrom u = 0 somewhat differently, as follows, nation m
be effected
somewhat
Solve the equation u = 0 for al, and let the result be differently.
u, = a1, then, differentiating, we find
du- = 0, in which a, does not appear.
Solve this equation for a,, and let the result be u,= a2,
and then differentiating, we find
dut 0, in which neither a1 nor a, appear;