which for any positive and integer value of n is always 0; and therefore the given series is simply equivalent to

which is evidently equal to the definite integral of (x).

Most of these theorems may, without considerably altering their forms, be applied to the definite integrals in finite differences, but it would too much extend this paper to insert them with their demonstrations: the following, for instance, is analogous to Theorem III. D' being used to denote a definite integral in finite differences, a and a+mh the limiting values of x, h being the increment of x or a, then we shall have

where

(a) denotes the error made by taking (n-1) terms of

the series for the true definite integral.

From this theorem and any others of the same nature, in which each term depends on the error made by taking a certain number of the preceding terms for the whole, many remarkable theorems with respect to the reproduction of functions, even when not continuous, may be deduced: for example,

* If there be n rows of quantities, the first row entirely arbitrary, any even row, as the 2 mth, formed so that the vertical difference.

* In series formed after this manner, it is a curious property that the difference of n+1 two terms in nth row, which are places distant =

terms in the row preceding.

n+1
2

x difference of two consecutive

of any term =

-m
n-2m

x the difference taken horizontally, and

in the odd rows (as the 2m+1th) the law be the same, except that

then the nth row will always be a repro-

Example for five rows:

The inverse problem, viz. given the definite integral of a function
in different circumstances, to find the form of the function itself,
is susceptible of various solutions, as those circumstances are
themselves varied: thus, if D, be the definite integral of any
function of x from x = a

-

1

h√1 to

x = a + h√-1,

x = a + 3 h √ -

a + 5 h √ −1,

* Thus if ▲, be the vertical, and ▲ the horizontal differences, and u1 any term in the
first line, u the corresponding in the second, &c.

π

and change a into x, we shall have the function itself by dividing 2S by h√1, but when the above series is divergent, it becomes troublesome to calculate the Analytical values of the divergent parts, to remove which difficulty we should attend to the values of the divergent series arising from differentiating successively the equation

thus o should be substituted for 1357, &c.

O for 13-33 + 5 - 73, &c.

&c.

Definite integrals may be applied to the expansion of functions when they contain negative powers of x, and they serve to determine the coefficients of such terms.

Thus, if we wish to find P, the coefficient of in ƒ(x), when such a term enters, we have

the upper line containing the positive, and the lower the negative,

supposing a, ß, a, b, &c. to be odd, for the terms containing even powers entirely disappear in the integration.

Hitherto we have taken the limits of the integral independent of r, the same results hold in general when x is used for a; but it is to be observed that when the limits contain x, (as x-h, x+h,) the integral remains a function of x, and therefore is capable of being integrated again; and the result may be called the second definite integral of the given function, which being integrated between the same limits, will give the third definite integral, and so on: we shall denote by D', D', D", &c. the successive definite integrals taken in this point of view: In the following theorem which is similar to Theorem I, the limits are x-h√1 and x+h√-1; it comprises that theorem as a particular case, and admits of a similar proof.

then shall (2h-1). u, be the value of the mth definite integral

of ƒ (x); n being made infinite in the value of u1.

This theorem by making m=-1 shews how to determine the quantity of which the definite integral is f (x), the differential coefficients are in this case of a negative order - n; n; that is, they represent the nth integrals of the quantities under them.

If we know the first, second, third, &c. definite integrals, we may find the function itself by the following formula.

THEOREM VII. If xh, x + h be the limits of integration, then shall

=

d1. D'ƒ (x) ̧ &c.

dx4

these general properties of definite integrals might be much more extended, but the consideration of them would extend this paper to an inconvenient length; but we may observe that in studying these properties we should divide the functions under the sign of integration into large classes possessed of some common property, such as vanishing when a, &c.: the results do not here possess altogether the generality of the former, but they are more remarkable, and even more useful in analysis. To this part of the subject, however, I shall not at present further allude, but conclude with observing that I have subjoined as an illustration of the use of the preceding methods,-the general resolution of Riccati's Equation, by means of definite integrals.