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(17) If the function be (tana)" the formula of reduction is (tana)"-o

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If the function be zi the formula of reduction is

(tana)

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(19) If the function be eo (cosa)" the formula of reduction is

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1 (20) If the function be 7 –, the formula of (a + b cosa)” reduction is b sin a

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CHAPTER III.

INTEGRATION OF DIFFERENTIAL FUNCTIONS OF TWO OR MORE WARIABLES.

Sect. 1. Functions of the first order.

IN order that a differential function of two variables of the first order, such as

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should be the differential of a function u, it is necessary that the condition

dP d Q

dy T do should exist. When this criterion of integrability holds good, we find

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The application of these formulae may be generally facilitated by observing that in the second term of the former it is only necessary to integrate the terms in Q which involve a only, and in the latter those terms of P which involve y only.

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then P = a + Q = 2 by;

d
therefore = 0 = −.
da,

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Integrating with respect to y, and observing that there is no term in P involving y only, we find

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Since P does not contain any term independent of v,

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dP then in-oxy- da

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