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forced attention to the subject. And even then, the discontinuous constant was only a new fundamental symbol, inserted in its proper place, in such form and manner as what I may call discontinuous investigations shewed to be necessary. In the method which I propose to explain, discontinuity not only appears in its proper place, but with its proper symbol.

When n terms of the series can be expressed in terms of n, the supposition n = ∞ will generally point out, in one way or another, whether any, and what, discontinuity exists. The method which I proceed to explain, while it depends for its strictness upon the passage from a finite to an infinite number of terms, does not require the actual expression of n terms as a condition of practicability.

n = 0.

As usual, let px, px, &c. represent the results of successive functional operations; the symbol or admits of two distinct characters, in the periodic and non-periodic cases. Either "xa, for a finite whole value of n, or for no whole value whatsoever, except in the extension In cases where pa is not periodic, it has this peculiarity; that px, whatever may be the value of a, will either increase (with x) without limit, or will, for successive whole values of n, give a series of approximations to m different limits which are severally roots of "x = x. I am speaking of positive or negative functions, and of real roots. With this proposition my only concern here is as to the case where or has one limit, in which case it evidently must give L=L, L being the limit in question. And this proposition is already well known in every part of mathematics. For instance, most direct methods of successive approximation depend upon the use of Taylor's Theorem, in a manner which will be recognized in the following particular case. If

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then the limit of successive operations gives a root of x = x; that is,

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But x, considered as the limit of "x, may be one root of px=x, for values of x intermediate to one set of limits, and another for another. For instance, let p be greater than unity, and let 8x = x2. Then we have

p° x = x2° = 0 when x <1 = 1 when a = 1 = << when x >1.

We might generalize the theorem, by a supposition which common algebra would admit. An equation of any degree, considered as one of a higher degree with evanescent terms, has infinite roots. In the common mode of speaking, we must say that pa is either infinite, or a root of $x = x. In that just alluded to, we should simply say that is a root of px = x.

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and is infinite for all values of x except - m, when a is greater than

1, and m, for all values of x, when a is less than 1.

=

is the root of $x = x.

But

m

It must, I suppose, be well known that successive approximation will not be vitiated by any error introduced into the approximate results, unless that error be so great that the process is made to tend towards another solution, or to increase without limit. For instance, in the solution of

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In

The value of x may be what we please at the commencement, or the obtained value may be altered; and the attainment of any degree of accuracy, though retarded, is not rendered impossible. a similar manner, a purely graphical process will lead to information upon the value of x in particular cases, such as with a little care may be made equivalent to demonstration. Let OA be a line equally inclined to OX and OY, rectangular axes to which the curve y = px is referred. Taking any point P1, which has the abscissa required, x,

proceed alternately from the curve to OA, parallel to the axis of x,

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and of OA to the curve parallel to the axis of y. The ordinates of the successive points of the curve P1, P2, P3, &c. are the values of px, p3x, p3x, &c. for the given value of x.

The general expression 4, is then one which requires a development of the following kind,

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where x1, x2, x3, &c. are specific quantities depending upon the function in question, and C, means 1 when a lies between a and b, and 0 for all other values, &c.

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where ax, ßx, yx, are given functions, not periodic, and pr is to be found. In my treatise on the Calculus of Functions already alluded to, it is shewn that the complete solution of the preceding is

φα= με + να . ξαν,

where μα is any particular solution, va any particular solution of px = yx pax, and fax the general solution of $x = pax.

Among the solutions of (1) is the series

Bx + yx. Bax + 7x. 7ax. ẞa2x + 7x. 7ax. ya2x ẞа3x + ...... obtained as follows:

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In which va = yx. vax = yx. yax....ya” X vaˆ+1 x,

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whence the expression in question becomes

μα γα.γαπ....γα"χ.μα"+12,

which cannot, as might appear at first sight, give a different value for every different value of ux: for, since two values of pa can only differ by some solution of ox yx.pax, the preceding expression is the same whatever value of or be adopted.

For an infinite number of terms of the preceding series, we have

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And the equation paya. pax, if va be one solution, can have no But others, except of the form vx.§ax.

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evidently satisfies px = yx pax; or if we take vaya. vax, we find the preceding product to be v÷vax. Consequently, the expression for the series in question is

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that for the sum of the series is discontinuous, and represented by

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I shall take the two instances given in my former paper, which will of course be the most satisfactory, as the difficulty was prior to the explanation. The first was the series

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aa x = x2° = ∞ C-∞, -1 + C-, + OC-1, +1 + C+1 + ∞ C11 ∞ ·

-8

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To explain the cases where ax = x, return to the expression for the series, which then becomes

μx {1-(x)},

giving in this case for x = 0, the value 0,

and for x = 1 the value 1.

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