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.

dy

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.

[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

satisfy the differential equation unless it make

dzy

da, as derived from the differential equation, infinite; that it may satisfy the

differential equation even if it make

day dx2

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

value of will satisfy the differential equation.

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

∞

dy

obtained for. Hence, any relation which satisfies the

dzy

given conditions and makes finite, will satisfy the diffe

rential equation.

And the same result holds even if be infinite, provided

Lastly, as when this result does not hold, the failure is due

day dx2

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

we see that the line in which the

[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 r 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 equation

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

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

c as variable parameters, shall seek to determine them so that the above value of y may satisfy the equation given.

These parameters, enabling us to satisfy r-1 arbitrary conditions, besides satisfying the differential equation, we may choose these so that

may

be the same in form as if c1, C2,... c, were constant. Now

from (3)

whence

be satisfied. Differentiating the first of these equations, we find in the same way that

be satisfied. And thus continuing we see that the system of

will hold true provided that the r-1 conditions