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XX.—A Demonstration of Lagrange's Rule for the Solution of a Linear Partial Differential Equation, with some Historical Remarks on Defective Demonstrations hitherto Current

Published online by Cambridge University Press:  17 January 2013

G. Chrystal
G. Chrystal
Affiliation:
Professor of Mathematics, University of Edinburgh.
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Extract

It seems strange that a principle so fundamental and so widely used as Lagrange's Rule for Solving a Linear Differential Equation should hitherto have been almost invariably provided with an inadequate demonstration. I noticed several years ago that the demonstrations in our current English text-books were apparently insufficient; but, as the method by which I treated Linear Partial Differential Equations in my lectures did not involve the use of them, it did not occur to me to analyse them closely with a view to discovering in what the exact nature of the defect consisted. The consideration of certain special cases recently led me to examine the matter more closely, and I was greatly surprised to find that most of the general demonstrations given are vitiated by a very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the examination of particular cases at all.

Type
Research Article
Information
Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 36 , Issue 2 , 1892 , pp. 551 - 562
Copyright
Copyright © Royal Society of Edinburgh 1892

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References

page 553 note * By Caucht's “Fundamental Theorem regarding the Existence of the Solution of a System of Differential Equations.” For a demonstration, see Kowalewski, Madame, Grelle's Journal, Bd. lxxx. (1875)Google Scholar.

page 555 note † Forgetfulness of this point seems to have been the real root of the many defective demonstrations that have been given of Lagrange's Rule, beginning with his own.

page 555 note ‡ Any particular non-singular solution may come from all the three forms (19) in the form f(u, v) = 0; or it may come from (191) in the form x – g(u, v) = 0; and from the other two in the form f(u, v) = 0; and so on. It should also be noticed that the given solution may be only a derivative of, and not wholly equivalent to, f(u, v) = 0. I insist on these elementary matters because they are so very much overlooked, especially by English mathematicians, the result having been a plentiful crop of fallacies.

page 557 note † Œuvres, t. iv. p. 82Google Scholar.

page 557 note † Œuvres, t. iv. p. 624Google Scholar.

page 557 note ‡ Œuvres, t. v. p. 543Google Scholar.

page 561 note * I have not been able to find this “alium locum.”