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

98 DEVELOPEMENT OF FUNCTIONS OF TWO VARIABLES.

Symbolical If we use a symbolical form similar to that in 110,
form of the
develope- this expansion may be expressed in the following manner,
ment and
of Taylor's {observing that r - is the coefficient of h'k"l-' in
Series. rr r(n - r)
(h + k)n expanded, viz:
f( A){ _(dah_+dk)l (d(h+dbl) &'}f( b
= edahf dbkf (ab).
Taylor's series expressed similarly may be thus written
f(a + h)= efh.a,
f ~(a) being supposed to be the same as f(a).
proofofthe tt 147. In the expansion of f(a + h, b + k) above obresult that hm kn
damdy tained, the coefficient of is d}'ddf(ab). Now if we
= dndc"y. FmFnn
had arrived at this developement by first putting b + k for
y and expanding, and then a+h for x and expanding again,
it is easy to see that the coefficient of m-n would have
rmFn
been dbdaf(ab), hence we must have
d] d f(ab) = (d' d f(ab),
a result which we obtained before in (106).