(8) This theorem may be extended to the product of any number of functions by means of the multinomial theorem, so that we have

(9) If n be negative in the theorem of Leibnitz,

(d) ̃"(uv) = ["da" (uv), and therefore

which is the general formula for integration by parts.

and

(10) In the last expression let u = 1; then

f" d x" u =

1.2...n-1

x2 dv

1 x3 d v

+

n

n+1 dx

−&c.) ;

ཉན

dv

dx

(11) In the theorem of Leibnitz let v = €"", then as

This result is of great use in the integration of linear differential equations.

(12) If we assume as before

Apply these equivalent operations to f(x), and indicate the successive differentials by accents affixed to the f; then

Therefore, writing these in an inverse order and effecting the operations indicated, we find

ƒ (x + nh) − f(x) = h[ƒ'(x) + ƒ′(x+h) + &c. + f' { x + (n − 1)h}]

+

h2

1.2

+ &c.

[ƒ'' (x) +ƒ" (x + h) + &c. + ƒ" {x + (n − 1)} h]

+ &c.

we may expand the factor (eds-1)-1 by means of Bernoulli's Numbers; (see Chap. V. Sect. IV. Ex. 9) when it

d

Applying these equivalent operations to

f(x) or ƒ'(x),

dx

multiplying by h and transposing, we have

(E1) f(x) = h {} + E" + &c. + E (~~ ))" + } E"" } ƒ'(x)

ƒ (x + nh) − ƒ (x) = h [} ƒ' (x) + ƒ' (x + h) + &c. +

The results in the two preceding examples are of great use in the approximate evaluation of definite integrals.

Poisson, Mémoires de l'Institut, 1823.

(14) Having given the transcendental equation
x = ce",

we can expand in terms of e by means of the logarithmic method of solving equations: for the root of the pre

ceding equation is the coefficient of in the expansion of -log (1)

This is easily found to be

Applying these equivalent operations to fdaf (x) we find

2h

ƒ (x) = f (x − h) + 1. ¿f'(x − 2h) +

(3h)2

1.2.3

f" (x - 3h) + &c.

This very remarkable theorem is given by Mr Murphy in the Philosophical Transactions.

(15) In a similar manner we may prove the more general theorem,

h2

f(x) = f (x − nh) + nhƒ' {x − (n + 1) h}

+n(n+2) ==,ƒ"{x−(n+2)h} +n (n+3)2

1.2

h3 1.2.3

ƒ''' {w'−(n+3) h}+&c.

(16) We know by the Calculus of angular functions that

Applying these equivalent operations to (a), we find

Français, Annales des Mathématiques, Vol. III. p. 252.

(17) In the same manner from the equation

14 = cos cos 20+ cos 30.

we obtain the theorem,

&c.,

$ (x) = $ (x + h) − † (x + 2h) + † (x + 3h) − &c.

+ p (x − h) − p (x − 2h) + p (x − 3h) − &c.

(18) Likewise by means of the equation

Ꮎ

= sin sin 20+ sin 30 - &c.

we find that

d

2

h + p (x) = 4 (x + h) − § 4 (x + 2h) + } p (x + 3h) − &c.

dx

- {p ( x − h) − } p (x − 2 h) + } † (x − 3h) − &c.}