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XIV. On the Equations of Loci traced upon the Surface of the Sphere, as expressed by Spherical Co-ordinates
Published online by Cambridge University Press: 17 January 2013
Extract
During the researches which I have made upon this subject since the composition of my former paper, printed in the first Part of this Volume, so many and so varied inquiries arising out of it, and essentially connected with it, have presented themselves, that I have been compelled to make a total change in the plan I had then laid down for the completion of the present section of my communication. I found that many subjects to which I there alluded might, with propriety, be omitted in the present case, as constituting little more, in reference to principles, than illustrations, however interesting they might be when viewed as properties of geometrical figures. On this ground, therefore, I have cancelled a considerable number of properties of the Spherical Conic Sections, and retained only one or two for the purpose of illustrating the method of discussing the properties of those curves. The remarks I intended to make upon the singular points of spherical curves, the geometrical signification of certain symbols, and other inquiries collateral to these, have grown into systems of themselves, or been attached to other dissertations (either wholly or partially completed), to which they seemed to be as closely allied as even to the present subject.
- Type
- Research Article
- Information
- Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 12 , Issue 2 , 1834 , pp. 379 - 428
- Copyright
- Copyright © Royal Society of Edinburgh 1834
References
page 383 note * So far as I have been able to discover, this formula is due to Euler, who employed it in a paper on “Spherical Trigonometry, by the method of maxima and minima,” in the Berlin Memoirs for 1753, p. 226. It is very remarkable how near Euler approached, in the passage here referred to, to the method of spherical co-ordinates, such as we have here developed; and yet he does not seem to have entertained the slightest notion that such a method of investigation was capable of general application. Had it once occurred to his mind, there is no doubt that Spherical Geometry would have been in a much more advanced state than it now is. Its principles have been fully developed, and its practice rendered familiar as a branch of elementary study.
page 391 note * Indeed, had we taken (θ − κ) for the angle made by the radius-vector of the current point in the circle and the first meridian, we should still eliminate the functions of θ − κ at step (14), as we have actually done with θ, and the result is therefore exactly the same.
page 405 note * It may here be remarked, that the varied form of the equation referred to was accidentally omitted in printing that article. Divide all the terms by cot λ‚ cot λ„ and we get for (VI. 7) the following:
page 416 note * Opp. tom ii. p. 491.
page 423 note * Mém. de l'Acad. des Scien. 1771, pp. 566–73. The method was also applied by him to space of three dimensions.
page 423 note † See the Mathematical Repository (No. 25.), for a discussion of these classes of quantities and their properties.
page 424 note * Crelle's Journal für der reine mid angewandte Mathematik, 8er. baud, 324 Seite.