By P. G. Tait

Excerpt from An effortless Treatise on Quaternions

To the 1st variation of this paintings, released in 1867, the next used to be prefixed: -

'The current paintings used to be began in 1859, whereas i used to be a Professor of arithmetic, and much extra prepared at Quaternion research than i will now faux to be. Had it been then accomplished I must have had technique of trying out its educating services, and of enhancing it, sooner than ebook, the place stumbled on poor in that respect.

'The tasks of one other Chair, and Sir W. Hamilton's want that my quantity usually are not seem until after the e-book of his parts, interrupted my already huge arrangements. I had labored out approximately the entire examples of Analytical Geometry in Todhunter's assortment, and that i had made quite a few actual purposes of the Calculus, specially to Crystallography, to Geometrical Optics, and to the Induction of Currents, as well as these on Kinematics, Electrodynamics, Fresnel's Wave floor, &c., that are reprinted within the current paintings from the Quarterly Mathematical magazine and the complaints of the Royal Society of Edinburgh.

'Sir W. Hamilton, whilst I observed him yet a number of days earlier than his demise, recommended me to organize my paintings once attainable, his being virtually prepared for e-book. He then expressed, extra strongly possibly than he had ever performed earlier than, his profound conviction of the significance of Quaternions to the growth of actual technological know-how; and his wish rather undemanding treatise at the topic may still quickly be published.

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**Additional info for An elementary treatise on quaternions**

**Sample text**

6. Kovalevskaya's Paper 31 equations whose fonn was similar to that given by Jacobi, namely, a*n ax' a=1 {3=1 a The coefficients G~J are assumed analytic around (0, ... ,0). Using the majorant technique of Cauchy and Weierstrass she proved the following theorem: A. If (x" ... , xr) 10, • • • , (Xl, ... , xr)nQ are n arbitrarily-chosen power series having a common region of convergence and if all of them are zero when Xl = 0, X2 = 0, ... *

3 is done exactly the same way. The logical gap in the argument will be of importance in the paper to be discussed in Chapter 6. 1. Introduction Kovalevskaya's work on Abelian integrals which reduce to elliptic integrals (1884) is the hardest of her papers to explain to a general audience since it assumes a detailed knowledge of the nineteenth-century work in this area. The subject of Abelian integrals in the nineteenth century was a vast corpus of results, many of which are not generally taught nowadays, even to specialists in algebraic functions.

Once it is recognized that convergence might be a problem, one must worry about the soundness of the method of undetermined coefficients. For example, even though we know that the system (1') generates a unique set of coefficients for given Co, it is not immediately clear-though it is not hard to prove-that the power series so generated has positive radius of convergence. This problem gave rise to papers by Cauchy and Weierstrass, and eventually by Kovalevskaya. Let us now look at these papers.