must be an exact differential, we must have

d (μ Q) d(μR) d(μR) __ d (μP)

dz

=

=

dz

=

= 0,

Multiplying these equations by P, Q, and R, respectively,

adding, and dividing by μ, we have

the same equation of condition which was before obtained.

When this equation is satisfied a particular form of the factor u will frequently suggest itself.

μ

In Ex. 3 the functions

(ay-bz)2' (cz - ax)2 (bx — cy)2

are integrating factors. In Ex. 4 the functions

1

(x + y + z)2

Equations not derivable from a single primitive.

8. To solve the equation Pdx + Qdy + Rdz = 0, when the equation of condition (8) is not satisfied.

In this case the solution consists of two simultaneous equations between x, y, z, one of which is perfectly arbitrary in form.

For representing an assumed arbitrary equation in the form f(x, y, z) = 0.....(20),

Now these two equations enabling us, when the form of f(x, y, z) is specified, to eliminate one of the variables and its differential, e. g. z and dz, from the equation given, permit us to reduce it to the form

Mdx+Ndy = 0,

Mand N being functions of x and y. Solving this, we obtain an equation involving an arbitrary constant, and this equation together with (20) will constitute a solution. By giving different forms to f(x, y, z) every possible solution may be obtained. What a solution thus found represents in geometrical construction is the drawing, on a particular surface, of a family of lines, each of which satisfies at every point the condition Pdx+Qdy + Rdz = 0. Now dx, dy, dz are proportional to the directing cosines of the tangent line. Hence the geometrical problem may be represented as that of drawing on a given surface a family of lines, in each of which the directing cosines coso, cos, cos x at any point shall satisfy the condition

Pcos + cos + R cos x=0.......................................... (21). Ex. Required the most general solution of the equation

which is consistent with the assumption that it shall represent a series of lines traced upon the ellipsoid whose equation is

It will be found that (a) does not satisfy the equation of condition (8).

indicating that the projections of the proposed family of lines will be a certain series of central conic sections.

If a = = b = =c=1 the proposed equation admits of a single primitive, viz. x2 + y2+z2 = 1. And any line traced on the surface of which this is the equation will satisfy the differential equation; for the equation (c) by which the lines are ordinarily determined is now reduced to an identity.

The above method of solution is due to Newton. Monge has however remarked that the general solution may be expressed by the equations (10) and (11) of Art. 4, viz. by the simultaneous system

μ

where is the integrating factor, and V the corresponding integral of the expression Pdx + Qdy. It is indeed shewn in

that Article that (22) does satisfy the differential equation provided that the condition (23) is satisfied. But there is no practical advantage in the employment of Monge's form. Applied to the problem of drawing on a given surface lines satisfying the condition expressed by the differential equation, it makes the determination of the arbitrary function (2) itself dependent on the solution of a differential equation. Thus in the example last considered we have, on giving to μ the value 2,

To apply this to the problem of drawing lines satisfying the conditions of the problem on the ellipsoid

it is necessary from the above three equations to eliminate From the second and third which here suffice,

x and y.

we have

whence

Therefore

The particular solution sought is therefore expressed by the equations (e) and (ƒ), which are together equivalent to the previous solution expressed by (b) and (d).

Total differential equations containing more than three variables.

9. It will suffice to make a few observations on the equation with four variables

Pdx+Qdy + Rdz + Tdt = 0......... ..(24), and to direct attention to the general analogy.

it is evident that, in order that it should be derivable from a single primitive, we must have

d

where refers to x not only as appearing independently,

dx

but also as implicitly involved in t; and so on for the rest.

which are the equations of condition of existence of a single complete primitive.

It is evident from the symmetry of the problem that the

It

must also hold here. But this is not a new condition. may be deduced from (26), by multiplying the respective equations of that system by R, P, and Q, and adding the results.