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VII. Sketch of a Method of Introducing Discontinuous Constants into the Arithmetical Expressions for Infinite Series, in cases where they admit of several Palues. In a Letter to the Rev. George Peacock, &c. &c. By AUGUSTU's DE MoRGAN, of Trinity College, Fellow of the Society, and Secretary of the Royal Astronomical Society.

[Read May 16, 1836.]

DEAR SIR,

Two years ago, I presented to the Society through yourself, the detail of some anomalies which I had observed to exist in certain series which I then produced. They arose out of investigations connected with Functions, and since published in my Treatise on that subject in the Encyclopaedia Metropolitana. But on further consideration, I find that I have not distinctly expressed the method by which the anomalies of the series in question may be reconciled, or rather by which the series may be so obtained that the

difficulties shall not appear.

I beg leave therefore, to request that you will lay the following view of the subject before the Society.

The assumption of a given form for a development amounted to an express exclusion of several considerations, which, so it happened, did not affect the results of ordinary operations, in cases where the form assumed was that of development in whole powers of a variable. Among the exclusions, was that of the possibility of a discontinuous constant, which was never considered, I believe, until the errors which the omission of it created in the inversion of periodic developments

Vol. VI. PART I. A A

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 = cc 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.

As usual, let par, poa, &c. represent the results of successive functional operations; the symbol qa admits of two distinct characters, in the periodic and non-periodic cases. Either p"a = a, for a finite whole value of n, or for no whole value whatsoever, except in the extension n = 0. In cases where pa is not periodic, it has this peculiarity : that p"a, whatever may be the value of a, will either increase (with a) without limit, or will, for successive whole values of n, give a series of approximations to m different limits which are severally roots of p"a = r. 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 p"a has one limit, in which case it evidently must give p 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 low = a ; that is,

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But poa, considered as the limit of p"w, may be one root of pa' = a, for values of a intermediate to one set of limits, and another for another. For instance, let p be greater than unity, and let pa: = wo. Then we have

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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 poa is either infinite, or a root of pa: = a. In that just alluded to, we should simply say that poa is a root of pa = a,

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and is infinite for all values of a except m, when a is greater than 1, and = - m, for all values of a, when a is less than 1. But — m

is the root of pa = a.

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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* = IT. by a continued fraction.

The value of a 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. In a similar manner, a purely graphical process will lead to information upon the value of p r 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 = pa is referred. Taking any point P, which has the abscissa required, a ,

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

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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 P., P., P., &c. are the values of ‘pa, poa, poa, &c. for the given value of a.

The general expression poa, is then one which requires a development of the following kind, a', C., + æ, C, 4 r. C., + ...

where w, w, w, &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.

Let 'pa' = {3a 4 yr. (paa...... (1), where aa, (3a, ya', are given functions, not periodic, and pris 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

‘pa = u + + va: , śaa, where ur is any particular solution, va: any particular solution of q, r = 'ya paw, and £aa the general solution of pa’ = (paa.

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obtained as follows:

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pa"w = 3a'a -- "ya"a. pa"'r, whence, for a finite number of terms,

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which cannot, as might appear at first sight, give a different value for every different value of ar: for, since two values of par can only differ by some solution of ‘pa = 'yar. ‘pair, the preceding expression is the same whatever value of 'pa' be adopted.

For an infinite number of terms of the preceding series, we have u a' Y ar. Yaar. ... ya’ ar. u ao ar. And the equation pa = 'ya. 'par, if war be one solution, can have no others, except of the form var. £ar. But 'ya'. 'Yaar. ... yao a

evidently satisfies pa = 'ya pair; or if we take va’ = yar. war, we find the preceding product to be var + va” w. Consequently, the expression

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