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The method employed by Lagrange may be used for the determination of the successive differentials of other functions.

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Therefore making these substitutions, and as
du , di

= cos x? + (-)! sin x?,
dx ldal

equating possible and inipossible parts, we have

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1.2

(27)

Let u =

+ 1

We might in this case expand the function and differentiate r times each term in the development, but as this

d' u would give expressed in an infinite series, the following method, due to Laplace*, is to be preferred. It is easily seen on effecting two or three differentiations that the form

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(3). 1.2.3 The product of (2) and (3) must be equal to the second side of (1), and as this last consists of a finite number of terms having positive indices, the terms in the product of (2) and (3) which contain negative indices must disappear of themselves. Hence taking the terms with positive indices only (6 + 1)?+14 Wet, d'

un

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SECT. 2. Functions of Two or more Variables. If u be a function of two variables w and y,

dr+su d'+s

dy'dx" dx'dys' Ex. (1) U = "y" ; 7 = 1, $ = 1,

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Generally, in a function of any number of variables, the order of differentiation is indifferent.

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