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NOTICES AND ABSTRACTS

OF

MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS.

UNIVERSITY

CALIFORNIA

NOTICES AND ABSTRACTS

OF

MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS.

MATHEMATICS AND PHYSICS.

MATHEMATICS.

On the Expressions for the Quotients which appear in the application of Sturm's Method to the discovery of the Real Roots of an Equation. By J. J. SYLVESTER, M.A., F.R.S., F.R.A.S.

MANY years ago I published expressions for the residues which appear in the application of the process of common measure to fx and f'x, and which constitute Sturm's auxiliary functions. These expressions are complete functions of the factors of fr and of differences of the roots of fr, and are therefore in effect functions of the factors exclusively, since the difference between any two roots may be expressed as the difference between two corresponding factors. Having found that in the practical applications of Sturm's theorem the quotients may be employed with advantage to replace the use of the residues, I have been led to consider their constitution; and having succeeded in expressing these quotients (which are of course linear functions of x) under a similar form to that of the residues, i. e. as complete functions of the factors and differences of the roots of far, I have pleasure in submitting the result to the notice of the Mathematical Section of the British Association,

Let h1 ha ha..... hn be the n roots of fr.

Let (abc. 1) in general denote the squared product of the differences of a, b, c....l.

....

Let Zi denote in general Σ((ho, ho, he;), where 01, 02,... Oi indicate any combination of i out of the n quantities a, b, c,...l, with the convention that Z=1, Z1=n; and let (i) denote {1+(-1)}, being zero when i is odd, and unity when then I find that the ith quotient Q, may be written under the form Q;=¿P2 • (x−h1)+;P2 (x−h2)+&c........... +¿P2(x−h„),

i is even;

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If we suppose

f'x

by means of the common measure process, to be expanded fx under the form of an improper continued fraction, the successive quotients will be the values of Q1 Q2.......... Qn above found,

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The numerators and denominators of these convergents will consequently also be functions of the factors exclusively. They are the quantities the sum of the products of which multiplied respectively by fr and f'r produce (to constant factors près) the residues. The denominators are expressible very simply in terms of the factors and the differences of the roots; and their values under such forms were published by me about the same time as the values of the residues in the Philosophical Magazine; the expression for the numerators is much more complicated, but is given in my paper, "The Syzygetic relations," &c., in the Philosophical Transactions.

By comparing the expression for any quotient with the expressions for the two residues from which it may be derived, we obtain the following remarkable identity :Z-Z1, i. e. (h ̧h2........ h;_;) × Σ(h2h2 ........ h¡)

i-1

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....

When the roots are all real, we have thus the product of one sum of squares by the product of another sum of squares (the number in each sum depending upon the arbitrary quantity i), brought under the form of a sum of a constant number (n) of squares, which in itself is an interesting theorem.

The expression above given for Qi leads to a remarkable relation between the f'x quotients and convergents to fx

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and let the successive convergents to this continued fraction be

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where the numerators and denominators are not supposed to undergo any reduc tions, but are retained in their crude forms as deduced from the law

N=Q; N,N-2

D=Q. D1--Di-g'

N(x) being 1, and D, (x) being Q, (x), then it may be deduced from the pub

lished results above adverted to that

Z1 Z3 Z

.....

D2(x)=

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Hence

Z+1

1

{%(hehe,... hoi) (−ho) (z−ho,) ...(h)}.

Σ{ ¿(ho, ho, • . • ho¡— 1) × (h ̧—ho ̧) (h ̧—ho2) .......... (h ̧—ho¿−1)}

=

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