will evidently not be satisfied unless

du

(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 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,

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.

d'y

dx2

PROP. A singular solution makes the general value of

deduced from the differential equation in its rational and

integral expression, to assume the form. 응.

[The demonstration is given in Chap. VIII. Art. 14.]

day
dx

This ambiguity of value of is evidently but an expres

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

from the differential equation, infinite; that it may satisfy the

day da

differential equation even if it make infinite; lastly, that

if it do not satisfy the differential equation, the curve it represents is a locus of points of infinite curvature, usually cusps, in the curves of complete primitives.

dy

dx

These are two equivalent expressions for the same value of

The question now is, under what circumstances this

and this by the rule for the evaluation of fractions of the form is equivalent to the value in either of its forms before

obtained for dy. Hence, any relation which satisfies the

dx

d'y

dx

given conditions and makes finite, will satisfy the differential equation.

And the same result holds even if dy be infinite, provided

da

Lastly, as when this result does not hold, the failure is due to the infinite value of

day

dx2

we see that the line in which the

locus of the proposed relation intersects the curves of primitives will be a locus of their points of infinite curvature.

[Transactions of the Cambridge Philosophical Society, Vol. IX. Part II.]

Legendre's Memoir of 1790 throws but little light upon the subject of this Chapter. But it exhibits the theory of the singular solutions of differential equations of the higher orders, both ordinary and partial, in a form of great beauty, and will be noticed in the proper places.

CHAPTER XXII.

ADDITIONS TO CHAPTER IX.

1. BY successive application of the second theorem of Chap. IX. Art. 13, a linear equation of the nth order may be reduced to one of the (nr)th order, if distinct integrals of what the given equation deprived of its second term would be are known.

The reduction may however be effected immediately by the method of the variation of parameters. In this and in most general investigations connected with differential equations great advantages in point of brevity and of the power of expression are gained by the employment of the symbol of summation, and of the language of determinants. I shall exemplify this here.

Suppose the given equation to be

and let y1, Y,...y, be r particular values of y, satisfying the

19

is a solution of the latter equation including these particular solutions. We shall represent this by

and regarding the quantities c1, C2,... C,, represented here by