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+(a2 + n2)1 cos 3

3.5 3.5.7.9

James Gregorie, Ib.

e"* cos næ = 1 + (a2 + n2)1 cos $ + (a2 + n2) cos 24

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Then

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1

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(9) Let

u = (1 + €')",

ƒ (0) = 2", ƒ'(0) = n2"1,

2-2

f" (0) = non-3 (n + 3), f (0) = n2′′ - 4 (n3 + 6n2 + 3n − 2);

ƒ" (0) = n2"−2 (n + 1),

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Maclaurin's Theorem may also be applied to the development of implicit functions, the differentiations being effected by the methods required in such cases.

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when a = 0, u=1, therefore ƒ'(0) = !.

Differentiating again,

2

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Since the given function is a quadratic in u it involves really two different functions of x, which in the development are given by means of the double sign.

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The possible root of this is 2, and if we take it, we find by the same method as in the last example the series

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The other series for u would be found by taking the impossible values of the cube root of 8.

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When a 0, u3 - au 0, which gives

=

=3

u = 0, u = ± a.

Taking the first of these values, we find the series

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expand y in terms of a.

When = 0, sin y = 0; therefore y = rπ, r being 0, or

any positive integer.

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In a similar manner we should find

ƒ""(0) = 2 sin a 3 - 4 (sin a)"},

and so on; therefore, substituting in Maclaurin's Theorem,

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(14) If u" log u = ax, expand u in terms of a.

When x = 0, one value of u is 1, as log 1 = 0; therefore taking f(0) = 1, we find

f'(0) = a, ƒ"(0) = − (2n − 1) a2, ƒ""' (0) = (3n − 1)o a3,

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As the calculation of the high differential coefficients of implicit functions is necessarily very tedious, this application of Maclaurin's Theorem is not of much use; and a better means of expanding implicit functions, is to be found in the Theorems of Lagrange and Laplace, to which we now proceed.

SECT. 3. Theorems of Lagrange and Laplace.

If y be given in an equation of the form

y = x + xp (y),

and if u = f (y), f and being any functions whatever, then u may be expanded in ascending powers of a by the theorem.

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This is Lagrange's Theorem. See Equations Numériques, Note XI; Mémoires de Berlin, 1768, p. 251.

The Theorem of Laplace is an extension of the preceding, made by assuming the given equation in y to be

d

y = F{≈ + xp(y)}.

Then if u = f (y), and if we put ƒ F(x) = f1 (*), and

__ƒ F(x) = ƒ{'(x), and pF(x) = p1(~),

dz

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Mémoires de l'Académie des Sciences, 1777, p. 99.

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