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419.] Let us investigate and define two or three laws to which symbols whether of operation or of quantity may be subject.

Let </), xfi be symbols of operation; and let u and v be symbols of quantity, and of subjects on which <f> and \fr are performed. And let us suppose <f>, u to be such that

(^Vr(tt) = f<p («)j (1)

that is, if the two operations indicated by <p and \jr are performed successively on w, one on the back of the other, let us assume the result to be the same whatever is the order in which they are performed: two such symbols of operation are said to be commutative, and to satisfy the law of commutation. Similarly, again, if <f>, \(r, \ are such that

<^X(") = ^X<*> («) = X#*(«)> (2) these symbols are commutative. Quantities multiplied into each other, satisfy this commutative law: thus if a, b, c are constants

oxhcxt( = ixcxflx« = cxhoxi(, (3)

Symbols of differentiation are also subject to the same law: thus, if u is a function of x and y,

dy dx ~~ dx dy'

and a similar theorem is true if u is a function of three or more variables. The form in which the law of commutation has been heretofore expressed as to differentiation is, "the order of differentiation is indifferent." See Art. 79. Trigonometrical functions are not generally subject to this law; thus, sin 2 x is not equivalent to 2 sin x; neither is sin tan-1 x equivalent to tan-1 sin x.

Again, let <^> be a symbol of operation, and a and v two symbols of quantity; and let <p be such that

<t>(u + v) = <f> (k) + <f> (v) J (5)

then the operation expressed by <f> is said to be distributive, and <f> is said to satisfy the distributive law. Similarly the number of subjects of the function may be n; and we may have

<f> («i + «a + +««) = <#> (Ml) + <f> + + <t> (mh)- (6)

Let us at present take only two subjects, as in (5); and let us operate on them again with the symbol if>; then, if we symbolize Price, Vol. i. 4 H

two such operations, performed successively on a subject by </>-, we have *«(« + ») = *{♦<« + »)}

= <M</> («) + + (»)}

= c/>2 («) + <p* (v); and if the operation is performed re times successively

f(M«) = f(«)+f(»). (7) Symbols of multiplication are in conformity with this law; thus, if a is a constant,

a(u + v) = au + av, a" (w + v) = a"u + a"v.

Symbols of differentiation are also subject to it; thus, if u and v are functions of x,

d , du dv

Tx{U + V)=dx + Tx* (8)

dx«K 'dxn ^ dx"'' trigonometrical operations are not subject to it; thus, sin (w +1) is not equivalent to sin « + sin v.

Constants and other symbols of multiplication are also subject to a law of notation; or rather a law of notation has been framed with respect to them to which other operative symbols may be subject, when they are repeated successively on the back of each other. The origin of this law which is analogically extended is the following; let a be a constant multiplied into u;

then ama"u = am+"u;

that is, if u is multiplied by a first n times, and then m times, the result is the same as if u had been multiplied m + n times by a. So if <j> is a symbol of operation which is performed on u first re times and then m times, and if the result of these successive operations is the same as if the operation of which <f> is the symbol had been performed m + n times, then

<t>mtf>n(u) = <f>" + "(M). (10)

this law of notation is called the iterative or the repetitive law. Evidently differentiation is a process subject to it; because it has been shewn in the preceding pages that dm d" _ d"'+n

dxTM dx"U d.vm +"W' *'

Now the calculus wherein these and other similar laws of operation, as distinct from symbols of quantity, are developed, is called the Calculus of Operations; and also is sometimes colled 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 0 be a distributive function of which u + v is the subject: and let it satisfy (5) and (7); let

a,, + ai <p + <h 4>* + +an<t>"

be an algebraical function of it; then, from (5) and (7), it is manifest that

{ao + «i<?> + a202 + +a„<t>n} (w + J>) =

{ao + ai<i(>-l-a2<J>2+ + an<f>n} u

+ {ao + ai<*> + «2<f>3+ +«»f}w; (12)

and if r symbolizes an algebraic function, then, also

r<p(u + v) = F<f)(w) + F <£(«); (13) and therefore as ^-differentiation is a distributive operation,

Again, ^-differentiation is subject to the commutative law: now suppose

d"u d"-lu d*~*u du _ ft

d^+P2d^~±Pn-i Tx+P"u = 0 <15>

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

dn du~1 d

dT«-*d^+ ±^"^+^ = 0; (16)

let the roots of this equation be Oi, a2,... a„; and let us resolve (16) into its factors; so that it is equivalent to

(£-».)(£-«■) (£-*) = «; ar,

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 M had been operated on by the whole symbol (16).

TM» (s-0(i-»). = (i-')(S-M

— — — 3 — 2m
~ dx* dx


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 assume

rf-1 =j; (18) so that (^) 1 = d-1 xdx = jdx, (19)

and "= d-"dx" = ^dx"; (20)

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 rath power; whereby we have


and adding the subject, we have

»"« = (£) + - (sS^) +... + (g) *■; (25)

which result is the same as (104), Art. 80.

If we use the notation of Art. 47, then Dm = dxu + dyu;

... D = dx + dv, and D» = dx» + n1 d9 + W("~1} <//+...+ (26)

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

D = dx + d, + d, + (27)

D" = (dx + d!l + dz+ )»; (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

d dn

emS = meme, .-. — em» = mnem«; (29)

and therefore if F represents an algebraical function

F (^) em» = emt> v(m): (30)

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


Next let us take the operating symbol ^e"'; that is, let us


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

of which the symbol is ;then

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