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Published online by Cambridge University Press: 17 January 2013
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.
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.”
