History of the Theory of Singular Solutions. 12. It is remarkable that while the theory of enveloping curves and surfaces was at once founded and developed by Leibnitz in 1692-4*, the corresponding theory of the singular solutions of differential equations has been of very slow growth. The existence of these solutions was first recognised in 1715 by Brook Taylor; it was scarcely more than recognised by Clairaut in 1734. Euler, in a special memoir, entitled Exposition de quelques Paradoxes dans le Calcul Integral, published in the Memoirs of the Academy of Berlin for 1756, first made them a direct object of investigation; but the foundations of their true theory were only laid in 1768 in his Institutiones Calculi Integralis. Laplace, Lagrange, Legendre, Poisson, Cauchy, and De Morgan have in various ways developed and extended that theory; but there has been so remarkable a want of unity and connexion in this long series of researches, that important portions of the theory appearing in a too isolated form have been neglected, forgotten, and rediscovered. I purpose here to give a brief account of what seems most characteristic, rather than of what is most original in their several researches; for the germs of nearly all subsequent discoveries on the subject are to be found in the great work of Euler. Taylor and Clairaut appear to have been led by accident to the noticing of singular solutions; the former while directly occupied on the solution of differential equations, the latter while discussing a remarkable class of problems relating to the connecting properties of different branches of the same curve. Taylor gave them the name singular, while Clairaut, and Euler too in his memoir, regarded them as a species of paradox, not merely from their non-inclusion in the general integral, but from the mode of their discovery through a process of differentiation. The memoir of Euler, though it sheds no light on the real nature of these solutions, contains * Acta Eruditorum, 1692, p. 168; 1694, p. 311. Opera, Tom. III. pp. 264, 296. Methodus Incrementorum, p. 26. Mémoires de l'Académie des Sciences, 1734, p. 209. an interesting theorem concerning their connexion with the form of the differential equation, viz. If this equation can be brought to the form. Vdz = Z(Pdx + Qdy), in which z is a function of x and y, and Z of z, then will Z=0 be a singular solution. In his Institutiones Calculi Integralis, Tom. I. p. 393, however, Euler gives a rule which is the counterpart of that of Cauchy. [See Chap. VIII. Art. 12.] He shews that if u=0 be a particular integral, and if the differential equation be reduced to the form The limits of integration are here supplied. The reasoning, which is not fully developed, is the following. From the transformed equation we have If this be satisfied by a solution involving x and y, and if that solution be a particular integral, then on putting for x its value in terms of u and integrating, the above equation will be satisfied by giving some particular constant value to C. But if the supposed particular integral be u= 0, then x and u being independent, we may perform the integration with respect to u as if x were constant. The resulting equation cannot be free from x unless C be infinite, and then it B. D. E. II. 3 du I will evidently not be satisfied unless (x, u) be infinite. We infer then that this is a necessary condition in order that u = 0 may be a particular integral. This is Euler's fundamental theorem, and from this, by means of an hypothesis agreeing with that of Poisson concerning the form of the transformed differential equation, he arrives at the condition [In the passage to which Professor Boole refers, Euler does not undertake to discuss the nature of any solution, but only of a solution of the form = constant. On his page 408 Euler proceeds to discuss the nature of any solution. Professor Boole seems to me to attribute too much to Euler. For the convenience of those who wish to examine the original, I will give the reference to the passages in the later editions of Euler's Institutiones Calculi Integralis: Vol. I. pages 343 and 355 of the edition of 1792; Vol. I. pages 342 and 354 of the edition of 1824.] Laplace in the Memoirs of the French Academy for 1772, p. 343, established the tests and shewed their respective uses. He established also the test which consists in the comparison of differential coefficients, and he supposes it universal. He adopts the hypothesis of his predecessors as to the forms of expansion, but with some recognition of its insufficiency. Lagrange in the Memoirs of the Academy of Berlin for 1774, p. 197, and 1779, p. 121, appears first to have developed the theory of singular solutions in its two forms of derivation from the complete primitive and derivation from the differential equation, and to have established the essential connexion of these. But supposing the differential equation to be expressible in the rational form F(x, y, p) = 0, and employing the differential coefficients of F(x, y, p) instead of those of p he was led to sacrifice rigour to symmetry. One of his results has often since been adopted as a test of singular solutions. It may be thus stated. PROP. A singular solution makes the general value of d'y deduced from the differential equation in its rational and integral expression, to assume the form. [The demonstration is given in Chap. VIII. Art. 14.] day This ambiguity of value of is evidently but an expres dx2 sion of the fact that the contact of a curve of the complete primitive and that of the singular solution is not in general of the second order. The result given in equation (5) of Chap. VIII. Art. 14 has also been adopted as the test of singular solutions. The researches of Poisson and Cauchy have already been noticed. It is certainly remarkable that the final test to which Cauchy's analysis led should be essentially the same as that which had been discovered by Euler so long before. Professor De Morgan has thrown an important light upon the nature of the conditions which are fulfilled by all singular solutions in the expression of which x and y are both involved. He has shewn that any relation between x and y which satisfies these conditions will day satisfy the differential equation unless it make as derived dx2, from the differential equation, infinite; that it may satisfy the dzy differential equation even if it make infinite; lastly, that dx |