There remains the case of x=0 when n is between

and. As this does not satisfy

d

dx

=∞, we infer that

it is a particular integral. To prove this we have

xn+1

c=yε n+1.

When x=0 this gives, since 1+ n is negative,

according as y is positive or negative. This is like Ex. 2 of Chap. VIII. Art. 8."

The remark made by Professor Boole in the above reply,

dp

dy

that if = leads to a solution which does not involve y nothing is to be inferred...is important. It corrects the statement put too strongly in Chap. VIII. Art. 7, “ All we can affirm is that if dp = gives a solution at all it will be a singular

solution."

dy

From Art. 8 onwards it seems assumed that a solution for

dy

which = 0 is always to count as a singular solution, even if de

it should coincide with a particular integral. This does not seem to have been quite the view of the former part of Chapter VIII. see Arts. 5 and 6 of the Chapter.

In Ex. 3 of Art. 9 we read, "the second is obviously a singular solution." This means that since we have a solution which makes infinite, we conclude that it is a singular

solution.

dp

dy

So in Ex. 5 of Art. 11 we read, "is evidently a singular solution," when it seems better to say, "and is therefore a singular solution."

4. The additional matter relating to Chapter VIII. begins with another example which was to be placed at the close of Art. 3 of that Chapter.]

This therefore is the singular solution and it satisfies both the tests, as both x and y are contained in its expression.

the first is not satisfied, the last two are satisfied.

The determination of c as a function of x by the solution

of the equation df (x, c)=0 is equivalent to determining

dc

what particular primitive has contact with the envelope at that point of the latter which corresponds to a given value of x.

One important remark yet remains. The elimination of c between a primitive y=f(x, c) and the derived equation dy =0, does not necessarily lead to a singular solution in the

dc

sense above explained. For it is possible that the derived equation

may neither on the one hand enable us to determine c as a function of x, so leading to a singular solution; nor, on the other hand, as an absolute constant, so leading to a particular primitive. Thus the particular primitive

whence c is if x be negative, and co if x be positive. It is a dependent constant. The resulting solution y=0 does not then represent an envelope of the curves of particular primitives, nor strictly one of those curves. It represents a curve formed of branches from two of them. It is most fitly characterized as a particular primitive marked by a singularity in the mode of its derivation from the complete pri

mitive.

All the foregoing observations and conclusions may be extended to the case of solutions derived from the condition

dy dc

5. We have seen that the equation = 0 may be satisfied by an absolutely constant value of c, so leading to a particular primitive and not a singular solution. In this case

(x+h, c) as well as (x, c) would vanish, and the numerator of (9), instead of being the difference of a finite and an infinite quantity, would be the difference of two infinite and equal quantities. [See Chap. VIII. Art. 8.] It would not therefore be infinite. Hence we conclude that would not become

dp

dy

infinite for a particular primitive in the strict sense of that

term, i. e. for a solution derived from the complete primitive by giving to c an absolutely constant value.

This is one point of contrast between the conditions

There is another not less important. As the numerator of (9) may become infinite not only when (x, c) = 0, but also when (x, c) = infinite, we see that a relation between

y and a which makes infinite will not necessarily satisfy

dp dy

the differential equation. On the other hand, it is not a particular primitive in the strict sense of that term.

as relating to the differential equation, with the same points of difference in the respective applications.

which becomes infinite when y = 0. As this involves y and satisfies the differential equation it is a singular solution.

To confirm this conclusion we may refer to the complete primitive

y= (x — c)",

which does not give y = 0 for any particular value of c.

Now let m be a positive constant less than 1. We have still =∞ when y = 0; but this value of y no longer satis

dp

dy

fies the differential equation. It is not a solution at all, nor would it result from the application of the condition

dy

dc

=0

to the complete primitive. The distinction of character of the two tests is here made manifest.

6. We may express the most important results of the foregoing investigations in the following theorem.

THEOREM. Every solution of a differential equation of the first order which is derived from the complete primitive by giving to c a variable value will, if it involve y in its expression, satisfy the condition

But relations satisfying these conditions will not necessarily be solutions of the differential equation.

In applying this theorem the following points must be carefully attended to.

1st. No conclusion can be drawn from the satisfying of

dp dy

the condition o when the relation in question does not contain y in its expression, nor from the satisfying of

when the relation in question does not involve x in its expression. For these conditions being respectively derived

In this

lar relation between x and y the indefinite form.