the last eight of these quantities being, as we have before shewn, expressible in terms of

We might proceed, by a continuation of the process of successive differentiation, to the transformation of partial differential equations in x, y, z, of any order whatever to equivalent ones in 0, 0, r. It is easily seen also that the same method of transformation may be extended to differential equations involving any number of independent variables whatever.

Ex. Transform the differential equation

where x and y are the independent variables, into one in which ✪ and & shall be independent variables, having given that

If we

involve n constants together with two variables x and y differentiate this equation successively n times, we shall obtain

n differential equations

Du = 0, D3u = 0, D3u 0,.... Du = 0,

=

involving the n constants, the variables x, y, and the 2n differentials

dx, d2x, d3x,. . . . . .d"x,

dy, d'y, d3y,. . . . . .d”y.

If we suppose de to be constant, then dex, d3x....d"x, will disappear from the equations. We shall thus have n + 1 equations involving n constants; and therefore, eliminating the constants, we shall arrive at a differential equation of the nth order in regard to the differentials of x and y, or, if x be the independent variable, of the nth order in regard to the differentials

of y.

Ex. Let

(x − a )2 + (y − b}2 = c2,

a, b, c, being constants: then

(x − a) dx + (y – b) dy

(x

a) d2x + (y b) d3y + dx2 + dy2 = 0,

(x − a) d3x + (y - b) d3y + 3dx d3x + 3dy d3y = 0.

From the first two of these differential equations there is

and therefore, from the third,

(dx2 + dy3) (d3x dy – dx d3y) + 3 (dx d2x + dy d3y) (dx d3y – d3x dy)=0.

We may also express the constants a, b, c, in terms of the second differentials of the variables.

In fact

Partial elimination of the Constants.

56. Instead of differentiating the equation

n times, suppose that we differentiate it successively only m times, m being some number less than n. Then we shall have m + 1 equations involving n constants: we may between these equations eliminate m constants, and shall thus obtain an equation of the mth order of differentials containing nm arbitrary constants. Since the m constants which we eliminate may be chosen arbitrary, it is evident that we may form as many equations of the order m, containing n - m constants, as there are combinations of n things taken m at a time: we may therefore obtain of such equations a number

n(n - 1) (n - 2)....(n - m + 1)

If we differentiate any one of these differential equations n - m times in succession, we shall have altogether n - m + 1 differen

tial equations involving n m constants. By the elimination of these n m constants we shall arrive at a differential equation of the nth order. It is important to remark that this final differential equation will always coincide identically with that obtained by the process of Art. (55). That such must be the case will be evident when it is considered that neither method of elimination involves any limitation of the generality of the variables and their differentials, and that accordingly the results must in both cases be perfectly general.

(x − a) d3x + (y - b) d3y + dx2 + dy3 = 0 . .. . (2).

1

Eliminating a between (1) and (2), we have

(y – b) (dxď3y – d3xdy) + dx (dx2 + dy3) = 0. . . .

Differentiating (3), we have

..(3).

(y - b) (dx d'y d3x dy) + dy (dx d3y - d2xdy) + d3x (dx2 + dy3) + 2dx (dx d2x + dy d3y) = 0,

or

(y - b) (dx d'y - dxdy) + 3dx (dx d'x + dy d3y) = 0..(4). Eliminating b between (3) and (4), we get

(dx2 + dy3) (d3xdy – dxd3y) + 3 (dxd3x + dyd3y) (dxd3y – d'xdy) = 0, a result coinciding with that obtained by the method of Art. (55).

Elimination of irrational, logarithmic, exponential, and circular Functions of known Functions.

...

57. Let u = f(x, y, c1, C2, C3,. . . .C„) = 0

be an equation between two variables x and y; where c,, C2, C3, .c1, are n irrational, logarithmic, exponential, or circular functions of s,, s2, 8,,....s, respectively, 8,, 82, 83,....s, being 'known functions of x and y. If we differentiate this equation successively n times, we shall obtain n differential equations

n

n

Du = 0, D3u = 0,

D'u

=

0,......D'u = 0,