operation, as distinct from symbols of quantity, are developed, is called the Calculus of Operations; and also is sometimes called the method of separation of symbols of operation from those of quantity its laws and results I proceed to develope, so far as they are applicable to differential calculus. Our operations thus far have been those of direct differentiation, that is, m and n in (11) are taken to be positive; for, although with certain limitations, the results will be true when the operations are inverse, that is, when m and n are negative, yet at present it is unnecessary to fix their limitations, and therefore I shall confine myself for the most part to direct processes.

420.] First I must shew that any algebraic function of a distributive symbol is also itself distributive; and therefore that an algebraic function of the symbol of differentiation is also itself distributive.

Let be a distributive function of which u+v is the subject: and let it satisfy (5) and (7); let

+{αo +α18+α2 p2 + ...... + a2p"} v ;

and if F symbolizes an algebraic function, then, also

F(u+v) = F $ (u) + F $ (v) ;

=

and therefore as x-differentiation is a distributive operation,

(12)

(13)

(14)

Again, x-differentiation is subject to the commutative law:

to be a differential expression; then, omitting the symbol of quantity, we have

let the roots of this equation be α1, ɑ2, an; and let us resolve

...

(16) into its factors; so that it is equivalent to

and if the operations indicated by these factorial symbols are successively, and one on the back of another, applied to the subject u, the result will be by the commutative law the same as if u had been operated on by the whole symbol (16).

It is not my intention to enter on the processes of general differentiation, when the indices are fractional, because the theory is too imperfect for an elementary treatise; but I shall extend some of the theorems to a few cases wherein the index of differentiation is negative; in which it will be convenient to use a distinctive symbol for this negative differentiation; and I shall

421.] First let us investigate certain forms which are taken by results proved in the preceding pages; by Art. 47,

so that if we omit the subject of the operating symbol,

that is, the operation expressed by D is equivalent to the two operations expressed in the right-hand member of the equation. As these operations are commutative, and as their symbols

are subject to the index law, we may raise both sides of (22) to the nth power; whereby we have

And if u is a function of many variables, then because

D = dx + dy + dz +

......"

D" = (dx + dy + dz + ................)" ;

(27)

(28)

the right-hand member of which must be expanded by the multinomial theorem, according to the process of Ex. 3, Art. 95 ; whereby the equivalent of D" will be obtained in terms of partial differentials and derived functions.

422.] We proceed now to the investigation of the results consequent on other symbols of operation beside those of simple differentiation.

Since

and I may observe that this result is true when n is negative.

d

Next let us take the operating symbol eme; that is, let us

do

suppose a subject u to be a function of 0, and to be multiplied by eme, and subsequently to be operated upon by the operation

and these equalities are true when n is negative; in which case,

using the notation of Art. 420, we have

From these expressions another theorem may be deduced. Since from (32) we have

in it let us substitute for m successively 0, −1, −2,... — (n−1); and let all these processes be performed successively on u; then we have

which and (30) and (37) are three fundamental theorems of this

kind of operating symbol.

423.] And they may by the following substitution take forms which are useful ir the solution of many differential equations.

Again, let the substitution (40) be made in (34) and (37); and

Also let the substitution (40) be made in (39); and we have

424.] By this last formula, differential equations, of which the differential terms are of the form ()u, that is, where a is equicrescent, and the power of x in the coefficient and the order of the derived-function are the same, may be transformed into others of which the coefficients shall be constant. Because,