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From these formulae. (sin bx)". (cos cx)" may be obtained, if (sin bx)" and (cos cx)" be expanded in multiple sines and cosines.

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series of

+ const.,

+const. if a<b:

cos x)

+ const.,

(a+b)(1 + cos x)

b+ a cos x

a+b cos x

=

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S = (a + b cos a)"

1

(n−1)(a2—b2) S +

20

b

+ const. if a > b.

log, (a + b cos x) + const.

α

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1

a+b cos x

(ab1-a,b) sin x

n-1

(n − 1)(a2 — b2)(a + b cos x)" =

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· 1)(aa1 — 2)(ab ̧

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· bb1)+(n − 2) (ab1 — a1b) cos x (a + b cos x)^-1

√(1+e.cos x) = Ax+4, sin x + 4, sin 2x + 4, sin 3x + &c.

2

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3

4

1... 4

e* + &c.

2.4

m (m − 1) (m − 2) 1.3

1. 2

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3

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(m-4) 1.3.5

m...

+

1...

5

2me A-2A1

(m + 2) e

(m-2) e A-23 A,

(m + 4) e

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2.4

e2

&c.}

·et + &c.

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(m − 1) e A1 — 23 ▲1⁄2 ̧
(m + 3) e

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If (1+e.cos x)-m=Ax+4, sin x + 4, sin 2x + &c.

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2

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+ 2 {k.cosx-k2. cos 2a+k3. cos 3x - &c.}

where k={1-(1- e°)}}.

(L. C. D. 449—66.)

APPROXIMATE VALUES OF INTEGRALS.

(23.) Let S-au represent the value of su, when x=a; then

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This series does not converge with sufficient rapidity,

unless b is very small: the following more convergent series

may however be obtained, by dividing the difference between the limits into n equal intervals;

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b2

b

u = = {(u),-a + (u)s=a+b+ &c. + (U)x=a+(n−1)b}

i

+ {dx=aU + dx=a+bu + &c. + d2=a+(n−1)bU }

+

1.2

b3

1.2.3

+ &c.

=

+

{d2 _qu+d2=a+bu + &c. + d2 =a+(n−1)b U }

= = { (U)x=a+b+ (U),=a+2 + &c. + (U)z=a+n}

b2

1.2

b3

{dx=a+bu+dx=a+2bU + &c. + dx=a+noU }

{d=a+bu + d2=a+2U + &c. + d2=a+nbu}

1.2.3 (d2

&c. b

α

= { { (U);=a+8 + &c. + (U)2=a+(n−1)) + § ⋅ (U)s=a + (U)s=a+n! }

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The values of u between the limits x=a, and x=a+nb determined from the first series are always less, and from the second, alternately greater and less than the true value. The third series, which is half the sum of the two first, is more convenient for calculation.

This method is inapplicable, if any value of a between the assigned limits renders any of the differential coefficients infinite. (L. C. D. 467-73; Tr. L. 209-13.)

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+

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1

1.2...(n-1)

(n − 1) (n − 2)

1 . 2

-3

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n- 1
·

ux

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1

uxn

a2¬3 ƒ ̧ua2 — &c. + (−1)”-1ƒ ̧uar-}·

(25.) Series independent of the integral sign.

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1

-2

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d2u +

du
U-

1.2 (n+2)

-&c.}

B

+ C1∞2¬1+C2xn−2+ &c. + cn;

C1, C2 &c. being arbitrary functions.

(L. C. D. 483-5.)

INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES.

(26.) If u is a function of ≈ and y, then

u=f&d ̧u+p(y),

an arbitrary function of y, which would have disappeared by differentiation, being added instead of an arbitrary constant.

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u=£x£y£zv+$1(x) + Þ2(y) + $3(x);

and so on, whatever may be the number of variables.

(L. D. C. 279-81.)

DIFFERENTIAL EQUATIONS.

any equation

(x,y) = 0 contains n independent

(27.) If constants, then

[1] n differential equations of the first order may be obtained, from each of which one of the n constants has disappeared:

[2] any one of these n equations will be of the same degree with respect to the differential coefficient, as the highest power of the constant which has disappeared:

n (n − 1) (n − 2)... (n − m + 1)

1. 2. 3

...

m

[3] there will be tial equations of the mth order, each involving n

constants:

differen

-m of the n

[4] there will be but one differential equation of the nth order, which contains no constant.

From any differential equation of the mth order m equations of the (m-1)th order may be obtained, in each of which one constant has been introduced. (L. D. C. 287-304.)

(28.) The criterion of integrability. If the equation

M+Nd,y=0

can be obtained by differentiation alone from the equation u = 0, then

and

d1M=d2N,
=S1N+S2(M—d2f,N).

ƒ2d,M=d ̧ƒ„M; by means of which either of the coefficients M, N, may be obtained from the other.

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