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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 for, mulæ 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,

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and therefore

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 f(+ 1) = f(w) + AS (@+imano Campo ' s (a)

+10.(a) 's(*) + &c., we have, by placing the function outside,

h d h2 l d ) 2 h Id 1 3 f (x + h)={1+

= {1+; dontja lãn +20G ) +&c.}f(x). 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 en in terms of hx, and as the symbol i s subject to the same laws of combination as the symbol x is supposed to be subject to in the demonstration of the exponential Theorem, we may consistently write the preceding equation under the form

f (x + h) = **f (x). As we shall have frequent occasion to speak of this operation of converting f (x') 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

f (x + h) = E, f (x) = 6*** 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

En f (x) = f (x + h), if k be another increment

ELE (2) = Ex f (x + h) = f (x + h + k) = En+x} (x), which is the index law. We may, therefore, put

f (x + 1) = 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

. du du d(u) =

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or, separating the symbol of operation from the subject,

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

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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

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(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, ...

d" u

dxa dyß dor... d" (u)=1.2...na

" dwa dyß dzy... 1.2...2.1.2...ß.1.2... y.. where a + B + y + &c. = n.

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

202 + A,an-+ A2-% + &c. + An-1W + A, is equivalent to

(x ) (« – 02) ... (x an); a,, , ... Qon being the roots of the expression equated to zero. It follows therefore that

is equal to

Como eres la ... o - - . )s

aj, aq ... an having the same meanings as before.

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

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

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- - ..) G- .) 6.- ...... - .) ..

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and Ni= (a, – az) (a, – az) ... (a, – an)' with similar expressions for N,, N3, &c. ... Nn.

It follows therefore that

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If we suppose r of the quantities a to be equal to each other, they will give rise to a series of p terms of the form


u where p receives all integer values from da do i to r. The value of the coefficient M, is easily found. For if we put

f(x) = (x – a)' ° (x),

$(x) when x = a.

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The results contained in the preceding four Examples are of great use in the Integration of Linear Differential Equations, and in the sequel I shall have frequent occasion to employ them. The theorem in Ex. 6 was first given by Mr George Boole of Lincoln, in the Cambridge Mathematical Journal, Vol. 11. p. 114.

(7) Binomial Theorem for differentials with respect to different functions. If u and v be two functions of x, then

du do

dodir Now if we accentuate the symbol of differentiation which applies to v to distinguish it from that which applies to u, we may write

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Affixing the index n to the symbols of operation on both sides,

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na moj

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or expanding the binomial on the second side by the theorem of Newton, we have

dau dv ds-lu (n-1) d'v dn-au V = V

d x" dr dra-1 " This is the theorem of Leibnitz, who arrived at it by induction for integer indices ; but it is true whether n be integer or fractional, positive or negative.


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