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

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.