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And as e

{ a = ( - ) ẞ } x

a x

= εa¿ {cos ẞx + (−)a sin ßæ},

and

{ a €

-(-)

= car {cos Br - (-) sin ẞæ}.

The sum of the two corresponding terms in (6) is

-ax

2€a1 (A cos ẞx + B sin ßæ) fdæ (e ̄ax cos ẞx. X)

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=

€ (C cos ẞ+ C' sin Ba) Ce cos (3x + a). (11)

If there be a number of equal pairs of impossible roots in the equation (3), the general expression for the value of y becomes so complicated as to be of little use, and it is therefore unnecessary to insert it here.

The preceding process may frequently be simplified in its application to particular cases, by means of the following considerations. The inverse operations are always reduced

to the sum of several of the form

be 1 or any positive integer.
a different effect according as
d
or descending powers of
considered to be

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dx

d

dx

+ a

1)

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X where may

Now this operation will have it is expanded in ascending that is, according as it is

d

dx

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X,

a +

dx

inasmuch as in the one case it will involve integrals, while

in the other it will involve differentials only. But a relation connects the two, for

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simple

-ax

= €

(a.

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' (Co + C1 ∞ + C2 ∞2 + &c. + C‚...‚œ”−1), and the former term

d

+

dx

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will give rise to a series of differentials which are always easily found, and which, when X is a rational and integral function of a of n dimensions, always breaks off at the (n+1)th term. But since each factor of the form

d

(a 2 +a)

dx

gives rise to a separate complementary function, while X is operated on by all in succession, it is sufficient to expand

{1}

in descending powers of X, without splitting it

into its binomial factors, and then to add the complementary functions corresponding to each of these factors.

If the function X be of the form e, the result of

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and then operate on e"

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mx

with each term separately, we find,

€”* = m' €TM*, that the series becomes

{A + Bm + Cm2 + Dm3 + &c.} "* = ƒ (m) eTM3.

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If the function X be of the form cos mæ

or sin mx,

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and if the operating function be ƒ (2) f

by the same method that, as

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()

2

it is easy to sce

m2 cos mx,

COS m x =

cos mx = ƒ (− m2) cos mx,

= ƒ (− m2) sin mx.

The preceding theory may be stated in the form of the following proposition: if the integral of an equation

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(2)

Let

+ ay = ε

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a1

5

a

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d

-1

y =

therefore

y=e'

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ay = 6TM1* cos ræ,

ar

a + m

(-a) (€TM* cos rx) = ea* [dx {em−a)= cos rx} ;

{(m- a) cosrx + r sin ræ}

+ Ce.

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therefore by the formula (11)

y = C cos n x + C' sin n x = C1 cos (nx + a).

The same result may be obtained by a different process, which is subjoined as it points out very distinctly the reason why these circular functions appear in the integral.

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It is indifferent in which order we perform the operations;

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