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Starting from these values and giving p successively all integer values from () upwards, we find

n

n' (1-2) cosnx = cos n(2r+1):1-3 (cosa) +

(cos.)'-&c.

1.2.3.1

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2

When n is an even integer the second line, being multiplied by the cosine of an odd multiple of vanishes, and the first line alone remains: when n is an odd integer the first line vanishes and the second line alone remains. When n is a fraction both lines must be retained, except for some particular values of n which cause the factor of one or other series to vanish.

(7) To expand sin nx in ascending powers of sin a'.

Proceeding in the same manner as in the last example, we find

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When n is an integer the first series always vanishes, and the second is positive or negative according as (n-1)r is even or odd.

When n is odd the second series terminates; when n is even it continues to infinity. When n is fractional both series coexist, except for particular values of r.

(8) To expand cos n v in ascending powers of sin x, and sin nu in ascending powers of cos x'.

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Proceeding as in the last two examples, we find

n?

° (no 9°) cos n x = cosnra {1 (sin x)' +

(sin x)' - &c.}

1.2

1.2.3. 4

- sin (n − 1) ra{n sin x

m (no – 1*)

(sin x)3 + &c.} 1.2.3

na

1.2.3

When n is an integer the second line always disappears, and the first series terminates when n is even, and does not terminate when n is odd. When n is fractional both series are retained, except for particular values of r.

n' (n-2) sinnx = sin n (2r+1){1-13 (cos.r)? +

(cos.) - &c. } 1.2.3.4

n (n° - 1) + sin (n − 1) (2r + 1) {n

{n cos x

(cos x)} + &c.} When n is an odd integer the first line, when n is even the second, alone remains; but when n is fractional both series are retained except for particular values of r. In no case do the series ever terminate.

For an exposition of the difficulties concerning these expansions, and the discussions to which they have given rise, the reader is referred to Poinsot's Memoir on Angular Sections, where the complete form of these expansions was first given.

(9)

To expand

in ascending powers of x.

€ - 1

If we were to endeavour to effect this by means of Maclaurin's Theorem, we should find that all the differential

0 coefficients take the form An artifice of Laplace* however

0 enables us to avoid this difficulty. Since

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odd powers of x above the first ; for if we assume

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and subtracting the latter from the former, x (1 – €")

- x = 2 {ax + azx2 + &c.},

6* and comparing the coefficients of like powers of x,

1

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Also it is easy to see that a, = 1; we may therefore assume

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Bi, B3, &c. being coefficients to be determined.

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the equation

(-)+ (2. - 1)

2n-1

But C22

B2-1

C.
1.2.3 (2n)
1
1
d

1

when x = 0, 22* 1.2 ... (2n - 1) and if in Ex. (27) of Chap. i1. Sect. 1, we make r = 2ń - 1, x = 0, d

1

2n (-)an [ - 221- 12n-1} + 1 24n

1

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2n-1

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Substituting these values, we find

(-)*+2n B1 =

120 {2n221 (24" - 1)

2 n (2n - 1) {321 - 1

1 2 -1} - &c.]. 1.2

2n

22n-1 +

+

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1

These coefficients B , B ,... B2-1, are of great use in the expansion of series, and bear the name of Bernoulli's numbers, having been first noticed by James Bernoulli in his posthumous work the Ars Conjectandi, p. 97; but the complete investigation of the law of their formation is due to Euler, Calc. Diff. Part II, Cap. v.

(10) To expand tan 0 by means of the numbers of Bernoulli

€(-)+ 20 tan 8

(-) 61-)* 20 + 1

1

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1 + 6

(-)28

The coefficient of Aan-' in the expansion of this function will be the same as that of an in the development of

E" + 1 multiplied by 22" (-)". . By what has preceded it appears, therefore, to be equal to

2?" (2?" – 1)
1.2... (2n)

B2n

2n-1

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(11) To expand cot by means of Bernoulli's numbers.

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Now the coefficient of 021-1 in this expression is the same

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