CHAPTER XV.

GENERAL THEOREMS IN THE DIFFERENTIAL CALCULUS.

In this chapter I shall collect those Theorems in the Differential Calculus which, depending only on the laws of combination of the symbols of differentiation, and not on the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols: but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first state briefly the theory on which it is founded.

There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are

The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is

indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a. That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew; but they are not confined to symbols of numbers; they apply also to the symbol used to denote differentiation. For if u be a function of two variables a and y, we have by known theorems in the Differential Calculus,

Also considering u and v as functions of r only,

The principal theorems in Algebra which depend on these laws, and which have therefore analogues in the Differential Calculus, are the Binomial Theorem with the great number of theorems-Exponential, Logarithmic, and others—which are derived from it; and the theorem of the decomposition of a multinomial of any order into simple factors with the various consequences which are deduced from it.

It is to be observed that in all the applications of this method to the Differential Calculus, a constant has the same laws of combination with the differentials that they have with each other, and therefore the theorems are true for complex symbols involving constants and symbols of differentiation. Also, there are two ways in which symbols of differentiation may differ from each other, either by having reference to different variables in the same function, or by having reference to different functions of the same variable, and this difference gives rise to two totally distinct series of theorems, as will be seen in the following examples.

It is worthy of remark, that the indices in the greater number of these theorems may be any whatever: I shall not however make any use of the interpretation of the formulæ when the indices of differentiation are fractional. It is easy to see that when they are negative they are equivalent to integrals of a corresponding positive degree: for by the law of indices,

this interpretation I shall frequently have occasion to use.

The principle of the method of the separation of symbols of operation from their subjects was first correctly given by Servois, in the Annales des Mathématiques, Vol. v. p. 93. Some very valuable researches on this subject by Mr Murphy will be found in the Philosophical Transactions for 1837.

(1) Taylor's Theorem. This theorem may be reduced into a very convenient shape by the separation of the symbols: for as

Now it is easily seen that the series of operations on the second side of the equation follows the law of the expansion of the exponential e" in terms of ha, and as the

d

symbol is subject to the same laws of combination as

the symbol is supposed to be subject to in the demonstration of the exponential Theorem, we may consistently write the preceding equation under the form

h

f(x + h) = e "* f (x),

As we shall have frequent occasion to speak of this operation of converting ƒ(a) into f(x + h) it will be convenient to denote it by a single symbol, and that which, following M. Servois, we shall employ is E; but as it is necessary to distinguish the value of the increment, we must attach to the symbol E the letter h. We might write therefore

nax f (x).

Farther consideration, however, shews us that the symbol h is subject to the index law, and may therefore be written as indices usually are. For as

E⋅f (x) = f (x + h),

if k be another increment

ExEn

Ex

E1E1 f(x) = E2 f (x + h) = f (x + h + k) = En+x f (x), which is the index law. We may, therefore, put

f(x + h) = E". f (x),

and throughout our operations consider h as an index.

(2) Binomial Theorem for differentials with respect to different variables,

If u be a function of two variables x and y, we have

or, separating the symbol of operation from the subject,

Affixing the general symbol n as an index to the operations on both sides of the equation, we have

Expanding the operation on the second side by the Binomial Theorem, since the demonstration of that theorem supposes only that the symbols are subject to the laws of combination before laid down, there results

(3)

In the same way, by means of the Multinomial Theorem, we may shew that if u be a function of any number of variables x, y, z...

d" (u)=1.2...nΣ.

dx" dy3 dz... 1.2... a. 1.2... ß.1.2... y...'

where a +B+ y + &c. = n.

(4) By the Theory of Equations it is shewn that the expression

a1, a2,... a, being the roots of the expression equated to zero. It follows therefore that

a1, a2... a, having the same meanings as before.

In this theorem it is necessary that none of the quantities A... A should contain u, x or y.

(5) If u be a function of one variable only, the preceding theorem becomes