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THE

QUARTERLY JOURNAL

OF

PURE AND APPLIED

MATHEMATICS.

EDITED BY

J. J. SYLVESTER, M.A., F.R.S.,

PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY,
WOOLWICH; AND

N. M. FERRERS, M.A.,

FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE:

ASSISTED BY

G. G. STOKES, M.A., F.R.S.,

LUCASIAN PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE ;

A. CAYLEY, M.A., F.R.S.,

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE; AND

M. HERMITE,

CORRESPONDING EDITOR IN PARIS.

ὅ τι οὐσία πρὸς γένεσιν, ἐπιστημὴ πρὸς πίστιν καὶ διάνοια πρὸς εἰκασίαν ἔστι.

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ON AN APPLICATION OF THE CALCULUS OF
OPERATIONS TO THE TRANSFORMATION
OF TRIGONOMETRIC SERIES.

By W. F. DONKIN.

THE greater part of the following paper was written a year ago. The subject was recalled to my recollection by Mr. Cayley's Note on the Expansion of the True Anomaly, in No. 7 of this Journal; and the paper is now offered as presenting what is perhaps a new view of the class of problems of which the Expansion of the True Anomaly is the most familiar example. The reasoning is founded on the properties of a series of which perhaps it might be said that it is practically more useful and theoretically more intractable than any other equally simple expression, insomuch that hardly any conclusion concerning it would command our unqualified assent without the confirmation derived from experiment.

SECTION I.

1. I use the term "Simple Periodic Series" to denote a series of sines and cosines of multiples of a variable. The most general form of such a series is therefore

Σ (A; cosix + B; sinix),

the summation extending to all integer values of i from - ∞ to +∞, including of course 0.

VOL. III.

B

Let S and C be used as functional symbols with the following significations:

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C(x)=1+2 (cosx + cos 2x + cos 3x+...),

the latter series being derived from the former by differentiation. More briefly we may write

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Putting a instead of x, I make not only the usual assumption that C(0-a) is always 0 unless -a be a multiple of 2π, but also the further assumption that the equation {$ (0) - $ (a)} C (0 − a) = 0 ...............................(3)

is always true provided (0) be such a function that

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when -a is a multiple of 27. This implies the assertion that when -a=2im the value of the left hand member of (3), which then takes the form 0x∞, is really 0; an assertion of which I can offer no proof which might not be objected to, though I believe its truth will be generally admitted, at least for ordinary forms of the function 4.

It is evident that if (0) be any simple periodic series, or any function of several such series, the required condition is fulfilled. The simplest instances are sinne, cosne, tanne, n being any integer.

In all that follows the symbol will be used to denote such functions only as satisfy the general form of the condition referred to; namely, 4(x+2iπ) = $(x). The conclusions however need not be considered as depending upon the truth of (3), as the principal theorem may be established on a different foundation. (See Art. 4).

2. If P be any function of 0, and n a positive integer, we have, as applied to any subject,

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In the case in which (2) is a simple periodic series, it can be proved by actual multiplication that (3) is identically true.

Now if for any reason we are entitled to put 0 for P in the right hand member of this formula, it is evident that the d" (P") whole series will disappear except the one term in don which does not involve any power of P as a factor; namely, ; so that, in such a case, we shall have

1.2.3...n

dP

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Now let P=4(0) - (a), and let the subject of operation be C(0-a); then since PC (0-a) = 0, we are entitled to put 0 for P on the right of (4), and consequently

d

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{$ (0) − $ (a)}" ( ~ )* C (0 − a) = (−)" 1.2..n {p′ (0)}" C (0 — a),

and if this transformation be applied to the terms on the right of the equation

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the series within brackets is reduced to

1 − p′ (0) + {p′ (0)}* — {p' (0)}3 +...,

or {1+ø'(0)}~1. Thus we obtain

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then C(0-a)= {1 + $′ (0)} C {0 + $ (0) — a − & (a)}....(5).

3. The series S(0-a) may be derived from C(0-a) by integrating every term of the latter from = a; hence, assuming that it is allowable to integrate each side of (5), we shall have

Theorem II. If S(x) =

then

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S(0 − a) = 8{0+ $ (0) − a − $ (a )} ......................... (6).

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As, however, the legitimacy of the last step may be questioned, it will be as well to deduce this equation independently.

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