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And by a process similar to that employed in the proof of Lagrange's Theorem, it may be shewn that

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dz d%z

In all these equations let x = 0; then z = a0) ^ = au = °3'

<Pz

-t-t= as,...; and y = /(ao); in whatever manner therefore the

flu.

dy d^y

values of ~, , ... given in (155), (156), (157), ... are composed of f(z),f\z), f"{z),... combined with ^, ~, i

in the same manner will these values, when x = 0, be composed of f(tio), /'(ao)> / "(flo), • • • combined with a1.O2.a3> • • • • From this

peculiarity we may deduce the following process; is the

differential of /(oo) on the supposition that dao = ax; [^jQ is

the second differential of /(ao) on the supposition that da<> = au

[dnv~\
is the «th differential of f(a0) on the

supposition that da0 = al5 o"ai = a%, da2 = a3,... dan_i = a„. In
this power of substituting new constants for the differentials of
other constants does Arbogast's method of derivation consist.
If therefore we replace the successive coefficients of the powers

of x in Maclaurin's Series, equation (13), Art. 58, by their values determined as above, we have

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