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630. ON AN EXPRESSION FOR 1 ±sin(2p+1)u IN TERMS OF sine. [From the Messenger of Mathematics, vol. v. (1876), pp. 7, 8.] White sin«--a>, then we have sin u = x, cos u - V(l - x*), sin 3m = 3,-c - 4?E», cos 3« = (1 - 4a'2) V(l - ."c2), sin 5» = 5« - 2(W 4- 16a5, cos 5« = (1 - 12a2 + 1G*4) VO -&c. &c. It is hence clear, that in general 1 - sin (2p + 1) u = (1 ± .t) ((1, xfl 1 + sin (2^ + I)it = (l + x) [(I, a;)?]2, where (1, a:)'' denotes a rational and integral function of x of the order p, and (1, - x~y the same function of - x for it is only in this manner that we can have cos2 (2p +1) u =(1 - x1) {(1, x*]*]1. We, in fact, find 1 -f sin ti = 1 + x, 1 - sin 3íí = (1 +x)(l - 2xf, 1 4 sin 5it = (1 + x) (1 + 2x - 4x*) 1 - sin 7m = (1 + x) (1 - 4t - 4^ + 8a.3)2, &c. and it thus appears that the form is 1 + (-)» sin (2;> + 1) u = (1 + c) 1(1, xy*.

Table of Contents

CONTENTS; [An Asterisk means that the paper is not printed in full]; 630 On an expression for 1 ±sm (2p+ i)it in terms of sine ; Messenger of Mathematics, t v (1876), pp 7, 8; 631 Synopsis of the theory of equations ; Messenger of Mathematics, t v (1876), pp 39-49; G32 On Aronhold's integration-formula ; Messenger of Mathematics, t v (1876), pp 8S-90; ^633 Note on Mr Martin's paper " On t/ie integrals of some differentials " ; Messenger of Mathematics, t v (1876), p 163; 634 Theorems in trigonometry and on partitions ; Messenger of Mathematics, t v (1876), p 1G4, p 188; 635 Note on the demonstration of Clairaut's theorem; Messenger of Mathematics, t v (1876), pp 166, 167; 63G On the theory of the singular solutions of differential equations of the first order ; Messenger of Mathematics, t vi, (1877), pp 23-27; 637 On 'a differential equation in the theory of elliptic functions; Messcngor of Mathematics, t, vi (1877), p 29; 638 On a q-formula leading to an expression for Et; Messenger of Mathomatics, t vi (1877), pp 63-66; C X; Tin ... Read more

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

Product Description
This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

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The book has no illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from the publisher's website (GeneralBooksClub.com). You can also preview excerpts of the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Original Publisher: The University Press; Publication date: 1898; Subjects: Mathematics; Mechanics, Analytic; ... Read more

Product Description
This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

Product Description
This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

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Arthur Cayley (1821--1895) was one of the most prolific and important mathematicians of the Victorian era. His influence still pervades modern mathematics, in group theory (Cayley's theorem), matrix algebra (the Cayley-Hamilton theorem), and invariant theory, where he made his most significant contributions. Yet Cayley's life has been overlooked by historians, who have lavished far more attention on lesser figures. Mathematician and biographer Tony Crilly, the world's leading authority on Cayley, rectifies this oversight with the first definitive account of his life.

Born in England, Cayley spent his childhood in St. Petersburg, where his father was a commercial agent. After returning to England in 1828, Cayley received a first-rate education. As an undergraduate at Trinity College in Cambridge, he was named "Senior Wrangler," the top mathematics student of his year. After graduating, he found himself at the vanguard of the revolution in British mathematics which included William Rowan Hamilton, George Boole, and James Joseph Sylvester. At the same time, needing a reliable income, he trained for the bar and became a barrister at Lincoln's Inn in 1849. Though a successful lawyer, Cayley devoted all his free time to mathematics and confirmed his reputation as one of the era's leading minds with a procession of brilliant articles on key aspects in pure mathematics. Only after 1863, when he was appointed to the Sadleirian Chair at Cambridge, could he fully pursue mathematical investigations, and he continued to publish influential papers until his death.

Comprehensive and elegantly composed, this biography makes clear the scope of Arthur Cayley's prodigious achievements, firmly enshrining him as the "Mathematician Laureate of the Victorian Age."

Customer Reviews (1)

A mathematical biography
You'll enjoy this book more if you're not mathematically knowledgeable because that's the audience it's written for. More than half the text deals with Cayley's mathematics but when the author feels it necessary to explain that a determinant is "a technical name given to a formula used to extract a single number from a spatial array", you can't expect deeper math topics to be treated with more than a superficial gloss. As a mathematician, I was disappointed in this book. ... Read more

Product Description
The book has no illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from the publisher's website (GeneralBooksClub.com). You can also preview excerpts of the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Volume: 2; Original Publisher: Cadell and Davis; Publication date: 1808; Subjects: Biography ... Read more

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This Book Is In French. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
The book has no illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from the publisher's website (GeneralBooksClub.com). You can also preview excerpts of the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Volume: 1; Original Publisher: Cadell and Davis; Publication date: 1808; Subjects: Biography ... Read more

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Originally published in 1920.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more