It will be observed that the previous general expression

น du

for

Sof(x, u)

becomes infinite if u =0 is a particular integral.

For then, F(x, 0) being a constant,

dF(x,0)
dx

vanishes, while F(x, u) – F(x, 0) does not vanish so long as u differs by however small a quantity from 0.

These propositions form the ground of the following Rule for the discrimination of singular solutions of the envelope species from all others.

11. RULE. The proposed solution being represented by u = 0, let the differential equation, transformed by making u and x the variables, be

in which U is either equal to f(x, u), or to ƒ(x, u) deprived of any factor which neither vanishes nor becomes infinite when u = = 0. If that integral tend to 0 with u the solution is

singular.

Ex. 1. Determine whether y=0 is a singular solution or particular integral of the differential equation

Here, since u=y, no preliminary transformation is needed.

which tends to 0 with y. Hence the solution is singular.

To verify this we observe that the complete primitive is

y=e,

and this cannot be reduced to y=0 by giving any constant value to c.

We have seen in Art. 7 that the test which is founded upon the comparison of differential coefficients does not suffice to characterize the above solution.

dy_ylogy is satisfied by y=0.

Ex. 2. The equation d

=

х

Is this solution singular or particular?

Here also no transformation is required. We have, reject

1

ing the factor which neither vanishes nor becomes infinite

when y = 0,

and this being infinite, however small y may be, may properly be said to tend to infinity as y tends to 0. The solution is therefore particular.

It will perhaps appear at first sight as if in the above example we ought to write

log y log log 0

when y is made equal to 0. But the course of the demonstration shews that the value of the definite integral must be first obtained on the hypothesis that u (in this case replaced by y) is finite, and then the limiting value which its expression approaches to, as u approaches to 0, be sought. And in this case, since for all finite values of u however small the integral is infinite, its limiting value is infinite.

The complete primitive in the above case is

y=e,

and the nature of the solution y=0 has already been discussed in Art. 4.

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

296.

*Acta Eruditorum, 1692, p. 168; 1694, p. 311. Opera, Tom. III. pp. 264,

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

From the

The limits of integration are here supplied. The reasoning, which is not fully developed, is the following. 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 3

B. D. E. II.

(x, u)

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 x = 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,