second differential coefficients of the dependent variable with respect to x and y.

I may perhaps at some future day resume the subject, together with an inquiry into the theory of the solution of the partial differential equation of this paper, when the conditions under which the auxiliary equations (I), (II) are supposed to be integrable are not satisfied.

9. NOTE. It may be desirable to establish directly the converse form of one of the results of Proposition IV. For this object we shall shew that the equation of the envelope of z=4(x, y, a, b, c).............

(1), where a, b, c are connected by any two conditions of the forms

↓ (a, b, c) = 0, x (a, b, c) = 0,

will satisfy a partial differential equation of the form

Rr+ Ss+ Tt+U (s2 — rt) = V ...............

in which also

=

-

аф do da do db, do dc

+

+

+

dx da dx db dx dc dx

аф do da, do db, do dc

dy

+

+

+

da dy db dy dc dy

and by the nature of an envelope these reduce to

(2),

Now since a, b, c are connected by two conditions, so that b and c are functions of x and y only as being functions of a, we have

This equation is of the general form (2). Its coefficients &c. are determinable as functions of x, y, z, p, q when dy, the form of the complete primitive (1) is given. For this purpose the complete primitive with the two derived equations (3) suffice.

Again, comparing (4) with (2) we have as the conditions of their equivalence

conditions which suppose R, S, T, U, V connected by the

relation

S2 — 4 (RT – UV) = 0.

CHAPTER XXX.

ADDITIONS TO CHAPTER XVII.

[THE present Chapter consists of additions to Chapter XVII. Art. 1 was intended to follow Chap. XVII. Art. 1.]

1. The theory of the solution of linear differential equations in a series flows very beautifully from their symbolical expression. It is usual in treating this subject to assume the form of the series, and deduce from the differential equation the law of its coefficients; but the symbolical form of the differential equation determines in reality the form of the solution as well as the law of derivation of its successive terms.

Let us begin with the binomial equation

fo (D) u − f (D) eo = = 0.

Operating on both sides with {f(D)}~1, we have u = $(D) €1o u = {ƒ.(D)}~10,

Now {f(D)-10 will be determined by the solution of a linear differential equation with constant coefficients, and will be necessarily of the form

AP+BQ+CR + ...,

in which A, B, C, ... are arbitrary constants, and P, Q, R, ... are functions of the independent variable.

We have then

{1 − (D) e1o} u = AP+BQ + CR+ ..., therefore u = {1- $ (D) e ̃oy ̃1 (AP+BQ + CR+ ...).

Now let us represent (D) ere by p; then

=

(1 + p + p2 + p3 + ...) (AP+BQ + CR + ...)

=A(1+p+ p2 + p3 + ...) P

+B(1+p+ p2 + p3 + .....) Q

+ C(1+p+ p2 + p3 +· .....) R

+...

Represent the first line of the above expression by u,, then since

u1 = A {P+c1o† (D+ r) P + €2ro † (D+ 2r) $ (D + r) P

Ε

+€3r0 $ (D+ 3r) $ (D + 2r) $ (D + r) P+.....},

in which it only remains to perform the operations indicated by (D+r), by 4 (D + 2r) $ (D+r), ... on the function P.

Let us in the first place suppose the symbolic function f(D) to be of the form (D − a) (D − b).....; then

Here Pe". Hence substituting in the above expression for u, and observing that f(D) ene = ƒ (n) eno, we find

u1 = Acao {1+$(a+r) eTMo + $ (a + 2r) $ (a + r) €2ro + .....}, or, since e = x,

u1 = Axa {1 + $ (a + r) x" + $(a + 2r). † (a + r) x2 + ...};