from which and the given equation y may be eliminated, and the resulting expression will be the integral required.

416.] Another form of differential equation of the first order and nth degree which admits of solution is

where F1, F2, ... F, are homogeneous functions of x and y; so that the equation admits of being put into the form

Let ytx; so that (235) takes the form f(y,t) = 0; we

and replacing y' by its value in terms of t which is given by f(y', t) = 0, the integral of the right-hand member of (236) may be found, and we shall thus have the general integral of 234. The following are examples of the process.

The geometrical interpretation of the given equation is "To find the equation to a plane curve such that the projection of its ordinate on the normal is equal to the abscissa."

We are now able to complete the theory of the relations between lengths of curves and the coordinates of their extreme points of which the more simple examples have been given in Art. 166.

Ex. 2. To find the curve the arc of which commencing at a given point is a mean proportional between the ordinate and twice the abscissa; that is, s2 = 2xy.

Ex. 3. To find the curve such that s2 = mx2+ny2.

(mx+nyy')2

.. 1+y'2 = mx2+ny2

.. (m+nt2)(1+y'2) = (m+nty')2;

whence we can easily find y' in terms of t; and by substituting in (236), we can determine the equation of the required curve. Ex. 4.2= x2 + y2;

dy
dx

t = y';

SECTION 9.-Partial Differential Equations of the First Order

and Higher Degree.

417. Partial differential equations of the first order and higher degree frequently offer themselves for solution in problems of

solid geometry; and it is incumbent on us to consider them so far as they are subject to integration; but here we are close on the boundaries of our knowledge; and it is often necessary for the complete investigation of functions satisfying given differential expressions to have recourse to considerations which belong to integral calculus as applied in geometry, mechanics, &c., and which are therefore beyond and extraneous to the fundamental principles of the pure science. For this reason we shall in the sequel omit some subjects which are to a certain extent within our present grasp; but which I believe it to be more advantageous for the student to defer to a future part of the course, so that he may have at his disposal all the materials which are available for the complete investigation. This course too is also historically preferable. Such equations as I allude to have arisen in physical investigations of light, heat, &c.; and they express properties referring to peculiar constitutions of the physical material of the theories from which these phænomena result. It is consequently with reference to these suppositions that they have been made subjects of inquiry, and it is in respect of these that their integrals become interpretable. Of some few partial differential equations of the first order and higher degree it is desirable however at once to seek the integrals.

418.] In the integration of these equations according to a received notation it is convenient to represent

dz

by p, and dx

dz

dy

by q; and let us suppose the equation which is proposed for integration to be of the form

f(x, y, z, p, q)

=

(237)

where is a dependent, and x and y are two independent variables; so that the integral is of the form z = F(x, y);

denotes the y-derived function of p on the assumption

dy

that x is constant; as however p may involve z which will vary

Also by means of the given differential equation q may be expressed in terms of x, y, z,p; wherein z is a function of a and y, and p is a function of x, y, z; and as we now require the x-derived function of q on the assumption that y is constant, we have

and substituting these values in (239), we have

dp

(dg) (dz) + (d) + {q − p (dg) } (dz) − (d2)—p(d2) = 0; (240)

dp

dz

dp which is a partial differential equation of the first order and degree; and consequently by the assumptions (110), Art. 387, we have

dx

dy 1

dz

dp

dq

dp

(241)

which is a system of three ordinary simultaneous equations, from which the integral of the given equation may be inferred according to the process of Art. 387.

It will be observed that from (241), we have

dx =

(da) dy,

dz = {q-p(d2)}dy;

.. dz = pdx+qdy;

so that while the system (241) is derived from the criterion of an exact differential it also secures that criterion.

Accordingly, if we can determine p by means of (241), and thence q by means of (237), we may substitute these values in (238) and thus determine z in terms of x and y. The determination of p will involve one arbitrary constant, viz. c1, and the integration of (238) will involve a second, viz. c2, which, by virtue of the argument of Art. 387, must be a function of the other constant.

The preceding process was originated by Lagrange, but was completed by Charpit, and now commonly bears his name. The following are examples in which it is applied.

which is the equation to a plane, the position of which depends on the arbitrary constant c1.

From the second and fourth of these terms, we have

(242)

which is an integral of the given equation.

Also from the third and fourth of (242) we have p =

therefore q =

21
; and

с

this is also another integral of the differential equation.