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be determined in terms of the preceding coefficients. But the labour of calculation from this expression may be diminished by observing that, as f(x) involves no odd powers of x except the first,

/'"(0) = =P-H0) = 0.

Hence in (126), if n is of the form 2r + l, we have

(2r+l)/"(0) + (^^)122r3(2r~1)/2-2(0) + ».

... + (2r+l)/'(0)+/(0) = 0, (127) by means of which any coefficient may be found; and we have

x I x 1 x* _ 1 x* 1 x6 noa\

2T + 6L2 30 E2^4 +42 1.2.3.4.5.6+ "( }

The values of /(0), f'(0), /"(0),... are commonly called the numbers of Bernoulli; and though they do not explicitly follow any palpably regular law, yet they are implicitly connected with each other by the formula (127). It is convenient to represent them by distinctive symbols; we will therefore substitute as follows:

B„=l, B!=— B2 = g> B3 = 0, B*=_ 30'

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88.] Of this theorem, it is expedient to give some examples.

Ex. 1. It is ^required to develope ^ ^ in terms of Bernoulli's numbers.

x x 2x

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89.] The sums of the powers of the reciprocals of the natural numbers may also be expressed as follows in terms of Bernoulli's numbers.

Since sin2 = 0, when z = 0, = ± it, = + 2it, = + Sit, ...; therefore sin z must be composed of factors of the form

z (z—it) (z + it) (z—2it) (z + 2it)...;

and therefore a relation of the following form must exist;

sins = **(l-£)(l-^)(l-g^)..., (134)

where k is an undetermined constant. But since by Lemma II, Art. 22, sin z = z, when z is an infinitesimal, therefore k = 1; and

«in. = .(l-^)(l-2^)(l-3^)

For z let us substitute itx; then

rin** = w*(l-£)(l-g)(l-f;) (185)

Of this equation let us take the logarithmic differential, and we

have f 1 2x 2x 2*

it cot itx = - - - ^-^ - Qt—s - (136)

and developing the several terms in the right-hand member, we have

it cotTM = --2x |—+ _ + _ + ... J

2xin~l jy^ + + + ... I .

In (132) let x be replaced by itx; so that

(137)

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Other theorems may be obtained by a similar process, if we take as the fundamental theorem the equation for the cosine which is analogous to that for the sine given in (135).

90.] It is unnecessary to enter at any length on the general subject of implicit functions of more than two variables, as the principles above explained and illustrated are applicable to all such cases. But as particular forms occur, and particular examples have to be solved in the sequel of our work, it is convenient to consider them at this point of the Treatise where they naturally occur; and I proceed to consider Lagrange's Theorem for the development of an implicit function of three variables, of the form y = z + x<j>(y), in ascending powers ofx.

Given that y = z + xtt>(y), in which equation y is an implicit function of two variables z and x, which are supposed to have no other relation to each other besides that given by this equation, so that they may vary independently of each other; it is required to determine f(y), another function of y, in ascending powers of x.

Let u =f(y), and therefore u is a function of x; whence, by Maclaurin's Series,

M = M«+ liJol + Lad. I* + Ldx->10TJ$ + (141>

using the notation of Art. 86 to indicate that particular values of the coefficients are to be taken, viz. when x = 0; that is, if

U = T(x), [w]o = P(0), jj^J =^(0), and so on. Hence we

have the following data:

«=/(y), y = z + x$(y); (142)

therefore when x = 0, y = z, and [m]0 =f(z);

now our first object is to determine the values of the quantities within the square brackets.

Calculating the partial derived-functions of (142), by considering y to vary in consequence of changes separately of x

and of z, that is, calculating and , we have

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