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1. [In Chapter 11. Art. 9, two methods are given for solving the differential equation

(ax + by + c) dx + (a'x + b'y + c) dy = 0.] But there exists another transformation by which the equation may be reduced to, (because it may be constructed from), an equation in which the variables are separated. Assume as this equation

(Ay' + C) doc' + (A'x' + C') dy' = 0...... (1) and let

ac' = x + my, y'= x + m y. It will be seen that in these equations united we have as many constants as in the original equation. Now on substituting in the assumed equation the values of ac' and y', and comparing with the equation given, we deduce a system of relations equivalent to the following, viz.: The quantities my, m, are roots of the quadratic

am? (b + a') m + b' = 0). The quantities A, A', C, C are determined by the system of equations

A + A' = a,

Am, + A'm, ra', Cm, + C'm,= c',




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C= cm

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from which we find

• a

- c
m, - me
- a'


C' m, - m2

m, m2 Now (1) gives on dividing by (A'd' + C') (Ay' + C) and integrating

A log (A’x' + C') + log (Ay' + C) =const.,

(A'x' + C") ^ (Ay' + C)ă = const.,
which on substitution and reduction gives
{(am, - a') (x + m,y) + cm, - c'}am, --

=const....... (2) {(am, - a') (x + m,y) +cm, - c'}am, -a' 2. Under certain circumstances the general solutions of differential equations of the first order fail

. This happens in the above example if my=m,, the solution then reducing to

1=const. The theory of the deduction of the true limiting form of the solution in such cases requires a distinct statement. Let the supposed general solution be represented by

u= C, C being the arbitrary constant and u a function of x, y, and constants which are not arbitrary. Suppose too that when one of these constants k assumes a particular value k, the function u reduces to a constant v. Then we have


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Now the second member being a function of an arbitrary constant is equivalent to an arbitrary constant and may be

replaced by C. The first member is a vanishing fraction, the

du limiting value of which is the brackets being used to denote that after the differentiation k is to be made equal to k. Hence the solution becomes

(dr), the brackets

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In applying this theory to the reduction of the general solution (2) in the case in which m, = mg, it must be observed that the numerator of the first member is the same function of my, x, y, as the denominator is of my, x, y; or attending solely to their functional character with respect to m,, mg, we may affirm that the numerator is the same function of m, as the denominator is of me. Representing these functions by 0 (m.), (m.) respectively, we have

φ (m)
$ (m2)

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But my, m, being roots of a quadratic equation may

be represented in the form

m, = m + k, me = m k,

the roots becoming equal when l = 0. Hence

$ (m + k)
$(m k)


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du dk

do (in + k)
$(m k)

(m k)
-(m + k)





1 - a

dd (m) dd (m)
20 (m)

dm dm therefore

{$(m) φ(mm)

= 2 dm log (m).


2 d

log (m) = C,

d 1
dm am

log {(am a') (x + my) + cm – c'}= C. 3. [The next Article seems to have been intended to appear in the enlarged form of Chap. II.; but I cannot discover what precise position it would have occupied. I conjecture that the above demonstration” refers to Chap. II. Arts. 2, 3; and I have accordingly supplied a reference to equation (3) of Chap. II.

I had myself drawn Professor Boole's attention to Chap. II. Arts. 2, 3.' The geometrical process of Chap. 11. Art. 3, appears to have been first given by D'Alembert in his Opuscules, Vol. iv. p. 255. D'Alembert calls it a demonstration; it seems to me only an illustration, at least in the brief form of the text: and that such was Cauchy's opinion may perhaps be inferred from the elaborate investigation given by Moigno, to which Professor Boole refers in Art. 5 of the present Chapter.

I had also drawn Professor Boole's attention to the statement at the end of Chap. 11. Art. 12, that only one arbitrary constant was involved. Accordingly Article 5 of the present Chapter developes this statement, and Article 4 seems intended to bear on the same subject.]

4. In the above demonstration the relation between y and

is regarded as one of pure magnitude, and the interpretation of the differential equation becomes a limiting case of that of the equation of finite differences (Eq. (3), Chap. 11.). But if we represent x and y by the rectangular co-ordinates

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