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multiplying the first of which by P, the second by Q, and the third by R, we have

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which condition must be satisfied, in order that (162) may admit of being made an exact differential by means of a multiplier : we shall return hereafter to the meaning of the necessity of this condition.

Now it is manifest that the three equations (165) are equivalent to any two of them together with (166): and if of these three integrals, involving three arbitrary constants, can be found, the most general integrating factor may be determined : if however we can integrate only one or only two, we may use the resulting expression as an integrating factor, although it may not be the most general.

Also from (165) in many cases, by various combinations, other forms of differential expressions may be found, whereby integrating factors may be determined. Thus one form may be obtained in the following manner: multiply the second of the group (165) by dz, and the third by dy, and then subtract the third from the second; and we have

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mined by the equation
$(C1, C2, C3) = 0;

(170) where • expresses an arbitrary function, and C1, C2, Cz are to be expressed by their equivalents determined as above. The most general form of the multiplier of course requires the integrals of all three equations; there is no method of finding the integrals of all in their above general forms; in many cases however, as the following examples shew, the integration is possible.

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which gives us a particular value of . And multiplying the given equation by it, we have zy dx Zx dy + y2 dz

= Du = 0;

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and multiplying the given equation by this, and integrating, we have u = ze + c';

(172) and therefore either this or (171) is an integral of the given equation ; and thus the general integral is

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the arbitrary functional symbol f including the arbitrary constant of integration.

398.] Equations (165) admit of combination into a more simple form when P, Q, R are homogeneous and of n dimensions : for multiplying the second of them by z, and the third by y, and subtracting, we have

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.D{u(Px +2y+R2)} = 0;

(173) Mo PX+QY +RZ' where c is an arbitrary constant: we subjoin an example in which the method is applied.

It is required to integrate the partial differential equation

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