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397.] We proceed now to a differential expression of three independent variables, of the form

Pdx+Qdy + Rdz

=

0;

(162)

where P, Q, R are functions of x, y, and z. Suppose μ to be a factor, by which, when multiplied, it becomes the exact differential of, say, the function,

and thus

u = F(x, y, z) = c; μPdx+μQdy +μRdz = Du = 0,

(163)

(164)

where

generally is a function of all the variables; then the conditions of its being an exact differential are, see equations (41),

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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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dy'

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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. . P { (du) dr + (du) dy + (du) dz } − (d) { rdx+Qdy + xd=}

dx

and therefore by (162),

dz

dx

μ { (dr) dx + (de) dy + (d) dz-dv};

edp = μ{(d)dx + (de) dy + (da) dz−dv};

= μ

dx

dx

};

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Ραμ

μ

.. log (μP) = == {(C) dx + (de) dy + (d) dz}+c1; (167)

similarly,

dx

log (49) = [ { { (dy)dx + (de) dy + (dy) dz}+c2; (168)

log (ux) = {(d) dr+ (d2) dy + (d) dz}+c; (169) f =

dz

dz

and the general form of the integrating multiplier will be determined by the equation

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where expresses an arbitrary function, and C1, C2, C3 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.

Ex. 1. zy dx-zxdy + y2 dz = 0;

in this case (166) is ―yz (x+2y) + xyz +2 y2z, which is equal to 0, and therefore the condition is satisfied; and from (167),

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which gives us a particular value of μ. And multiplying the given equation by it, we have

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and multiplying the given equation by this, and integrating, we have

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

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Ex. 2. (bz-cy) dx + (cx — az) dy + (ay — bx) dz = 0;

in this case (166) becomes

-2a (bz-cy)-2b (cx-az)-2c (ay-bx),

which is equal to 0, and therefore the condition is satisfied. The equations for determining μ become

μ (bz-cy)2 = 1,

μ (ex-az)2= C2,

μ (ay-bx)2=C3;

and therefore any value of μ which will satisfy the equation

• {μ (bz —cy)2, μ (cx — az)2, μ (ay—bx)2 }

= = 0,

may be used as a multiplier to render the given equation an exact differential.

Let us however take one of its particular forms, say the first of the group, and we have

(bz-cy) dx+(cx-az) dy + (ay-bx) dz

(bz-cy)2

= Du = 0;

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By taking the other values of μ we might obtain other, although equivalent, values of u; and thus the most general form of the integral is

u = F

сх -az bz-cy

= 0.

Ex. 3. (y2+yz+z2) dx + (z2 + zx + x2) dy + (x2 + xy + y2) dz=0. The condition (166) is satisfied; and to determine μ let us have recourse to first principles:

d

d

dz · μ (z2 + zx + x2) = dy •μ (x2 + xy + y2) ;

− (x2 + xz + z2) (du) = 2 μ(z − y) ;

(x2+ xy + y2)
́y3) (du) — (x2 + xz + z2)

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dy

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dz

αμ
2μ(z-y)

y2 + xy — xz — 22

dx + dy + dz

x + y + z

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=

dp: 2μ

=

log (x + y + z)2;

C1

(x + y + z) 3 3

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and multiplying the given equation by this we have (y2+yz+z2) dx + (z2 + zx + x2) dy + (x2+xy+y2) dz

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;

= Du = 0;

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and thus the general integral becomes

u = F

( y z + zx + xy) = 0,

x + y + z

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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and therefore multiplying by dx, dy, dz, and adding,

D{μ(Px+QY+Rz)} − μ (1+n) (P dx+Q dy + Rdz)

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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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