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Published online by Cambridge University Press: 06 July 2012
The theory of the singular solutions of differential equations of the first order, even in the interesting and suggestive form due to Professor Cayley (Mess. Math., ii., 1872), as given in English text-books, is defective, inasmuch as it gives no indication as to what are normal and what are abnormal phenomena. Moreover, Cayley added an appendix to his theory regarding the circumstances under which a singular solution exists, which is misleading so far as the theory of differential equations is concerned, if not altogether erroneous.
The main purpose of the following notes is to throw light on the point last mentioned by means of a number of examples. I have also taken the opportunity to furnish simple demonstrations of several well-known theorems regarding the p-discriminant which do not find a place in the current English text-books.
page 809 note * It is easy to see in another way that φx + pφy = 0 is simply the condition that the p-discriminant is an integral of the equation: for at any point of the p-discriminant dx : dy is given by
Since φr = 0, the first of these equations gives
The condition that the p-discriminant be a solution is p = dy/dx; hence,
page 810 note * It may be of interest to note that this is the condition that the function
obtained by retaining only such terms of the characteristic of the differential equation as are required to determine an accurate first approximation to the p-discriminant, is decomposable into factors which are integral and linear in x, y, p.
page 813 note * The general existence of the cusp-locus of the integral family of a differential equation of the first order was indicated as early as 1851 by De Morgan (Camb. Phil. Trans., vol. ix. pt. 2, p. 113). The earliest absolutely explicit statement of the theorem seems to have been made by Darboux (Comptes Rendus, t. lxx. p. 1331; also t. lxxi. p. 267, 1870). In an extremely interesting paper in the Bulletin des Sciences Mathematiques, &c, t. iv., 1873, p. 158, Darboux establishes most of the propositions above given. It is surprising that Darboux's work does not seem to have attracted the notice of Cayley. Reference may also be made to Clebsch, Mathematische Annalen, Bd. vi. p. 211, 1873; and to Clebsch's theory of “Connexes,” Vorlesungen uber Geometrie, Bd. i. p. 1014 et seqq. We have thought it worth while to deduce these results throughout by the approximative method first employed by Briot and Bouquet, because this method is a general one, applicable to the discrimination of special cases, such as arise when an envelope is also a cusp-locus or a tac-locus, &c.; and because this method is little used by English mathematicians.
page 820 note * The curves are unicursal quartics : by considering the intersection with the parabola 3y = μx(x−c) we find
page 820 note † There is an oversight in the interpretation of No. (xx).
page 820 note ‡ The two equations are equivalent to
that is to
which gave x = 0, y = 0 six times.