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d' (uv), Therefore, expanding by the Theorem of Leibnitz,

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(18)

Let u v = 6:44 X, X being any function of w.

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This result, when generalized, is of great importance in the solution of Differential Equations.

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By developing in a different manner a more convenient formula may be obtained :

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Developing w{(1 + " 19 + det va?" by the binomial

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and the pith differential of u" is the coefficient of h in this expansion multiplied by 1.2.... Now expanding each term by the binomial theorem, we have for the coefficient of

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The goth differential of this function may be found as in the last example, but the following method gives it under a form which is more convenient in practice;

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Hence we have

(d). 1 .(-)r(r – 1...2.1 sin (r + 1)
(dx) a + me

a

rti

(a + x)* Liouville, Jour. de l'Ecole Polytechnique, Cab. 21, p. 157.

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Liouville, Ib. p. 156. These results are useful in the theory of definite integrals.

In the following examples the functions are reduced to the required forms by differentiation in the same way as in Ex. 11.

2 du 1

(22) Let u= (1 -moi dan - 11

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