2ndly. Legendre's tests for differential equations of the higher orders are in kind and effect analogous to the tests

for differential equations of the first order. They enable us to decide whether a solution possesses singularity, not whether it possesses the envelope species of singularity. The completion of Legendre's theory would consist in the discovery of those further tests dependent upon integration which correspond to the test of Euler and Cauchy for differential equations of the first order.

CHAPTER XXIV.

ADDITIONS TO CHAPTER XIV.

[Art. 1 was intended to follow Chap. XIV. Art. 2.]

1. As the condition of dependence of functions of two variables is of fundamental importance in connexion with the theory of ordinary differential equations, so the generalized condition of dependence of functions of any number of variables forms a fundamental part of the theory of partial differential equations. This is contained in the following proposition.

...

PROP. I. If u1, u,... un are functions of x1, x2, x, but are as such so related that some one of them is expressible as a function of the others, or more generally that there exists among them some identical equation of the form

F(u1, U2, ... U2) = 0,....

..(1), so that as functions of x, x,,... x, they are not mutually independent, then, adopting the notation of determinants, the condition

is identically satisfied. Conversely, if the above condition be identically satisfied, the functions u,, u,,... u, are not mutually independent in the sense above explained.

First let it be noticed that the Proposition is but a generalization of that of Chap. II. Supposing U and u to be two functions of x and y, the condition of their dependence is affirmed to be

i. e. it is the result of eliminating dx, dy, from the equations

We proceed to the general demonstration.

Let the first member of (1), considered as a function of И19 u1, u2,... u be represented for brevity by F; then differentiating, we have

dF

du

n-1

from which it follows that if du,, du,,... du, are equal to 0, then is du, equal to 0; or, since u1, u2,... u, are functions of X1, X27 xn, that if

Thus the last n equations, linear with respect to

dx1, dx2,... dx

are not independent, and therefore by the theory of linear equations the determinant of the system vanishes identically. Now this is expressed by the condition (2).

Ип

It remains to prove the converse, viz. that if the condition (2) be identically satisfied, the functions u1, u,, ... u, will not be mutually independent.

First, the n-1 functions u1, u,,... U are either mutually independent or not mutually independent.

If not, then the n functions u,, u,,...u are not mutually independent, and the Proposition to be proved is granted.

If u1, u,,... U are mutually independent as functions of x1,x,..., they may be made to take the place of n-1 of these quantities, e. g. x, x,... in the expressing of u Xn-1 un, i. e. we may, by means of the expressions for u1, U2, ... Un-11 eliminate from that of u, the quantities x, x,... 12 and so express u, as a function of u,, u,,... U and x. Suppose Un-1 this done, then the system (3), (4) will be converted into

du, = 0, du,= 0, .... dun-1= 0,

dun duns + dxn
dun

dun du1 + du2
dun

du... +

dun-s

du

dx

= 0.

Now, the determinant (2) vanishing, the equations of the linear system (3), (4) are not independent; therefore those of the transformed system, as written above, are not independent; therefore the last equation of that system must be a consequence of the others which manifestly are independent. But from the form of that last equation we see that such cannot be the case unless we have

n-1

which implies that u, is a function of u,, u,,... u- merely. Hence the functions u1, u2, u are not independent, as was

to be shewn.

The first member of the equation of condition (2) is commonly called the functional determinant of u1, u,,... u, with respect to x1, x2,.... The proposition may therefore be expressed as follows.

The condition of dependence or independence of any system of functions of as many variables is the vanishing or non-vanishing of the functional determinant of the system.

On account of the great importance of this proposition it is desirable to illustrate it by an example.

Ex. Are the functions

x+2y+z, x-2y+3z, 2xy - xz + 4yz — 2z3

mutually independent or not?

The equation of condition is

− 4 (− x + 4y − 4z) + 8 (2ỷ − z) — 2 (2x + 4z) = 0,

which is identically satisfied. Hence the functions are dependent. In fact, representing them by u, v, w, we have 4w= u2 - v2.

[Art. 2 was intended to follow Chap. XIV. Art. 4.]

2. As it has been shewn that a primitive

u = (v)

leads to a linear partial differential equation of the form

.(2),