Failure of the Development of f(x + h, y + k) by Taylor's Theorem.

93. The development of f(x + h, y + k), given in the preceding article, fails for particular values of x and y, whenever f or any of its partial differential coefficients becomes infinite; this failure being consequent upon the failure of Stirling's theorem applied to the expansion of (a). It will likewise cease to be applicable for particular values of x and y, which render f or its partial differential coefficients essentially indeterminate.

Limits and Remainders of the Development of f(x + h, y + k).

94. Let R, be the value of the remainder which must complete the series (1) of Art. 92, supposing this series to be stopped at the end of the (r+ 1)th term; then, by Art. (90),

0, 01, denoting certain unknown numbers comprised between O and 1. Hence, if we stop at the term

of the series for the development of f(x+h, y + k), we must add the complementary remainder Pri where

The numerical fractions 0, 01, which enter into these formulæ, being unknown, we cannot employ the formulæ for the actual computation of p,: they serve only in fact to determine limits between which the value of Pr must lie.

The value of R,' is equal to the definite integral

whence, for the actual determination of p,, we have

By analogous reasoning we might shew also, as an equivalent formula, that

In precisely the same way we might investigate symbolical formulæ for the remainder in the development of a function f(x + h, y + k, z + l, ...), x, y, z, ... being any number of variables.

Example of the Application of Taylor's Theorem for two

Variables.

95. Let f(x, y) = 0 be the equation to a curve, f(x, y) being a rational function of x and y; and let it be proposed to transform this equation into an equivalent one for a new origin (a, B).

Putting ax, B+y, for x, y, we have, for the transformed equation, 0 = f(a + x, B+ y),

or, by Taylor's theorem, the dimensions of the proposed equa

then

0 = Ax2 + By2 + 2Cxy + 2A'x + 2By + C':
ƒ = Aa3 + Bß32 + 2 Caß + 2A'a + 2B′ß + C',

the transformed equation will therefore be

0 = Aa2 + Bẞ2 + 2 Caß + 2A'a + 2B'ß + C'
+ 2x (Aa + Cß + A') + 2y (Bß + Ca + B')
+ Ax2 + By2 + 2Cxy.

Stirling's Theorem applied to Functions of two Variables.

96. If, in the development of ƒ (x + h, y + k) by Taylor's theorem, we substitute 0 and 0' in place of x and y, where O and O' are used to denote zero values respectively of x and y, and then replace h, k, by x, y, respectively, we have

which constitutes an extension of Stirling's theorem to functions of two variables.

The expressions for the limits and remainder may be obtained at once from those for the development of f(x+h, y + k), by first putting x = 0, y = 0′, and then replacing h, k, respectively by x, y.

Lagrange's Formula for the Development of Implicit Functions. 97. Suppose that

y being an implicit function of x and z by virtue of the equation

Ꮖ

The object of Lagrange's formula, which we proceed to investigate, is to enable us to develop u in a series arranged by ascending powers of x, and which does not involve y.

If (y) be any function of y, y being a function of x and z, then

¥ (y). {¥ (y) • dx)

dz

for it is plain that each of these expressions is equal to

(3);

Differentiating (2), considering z constant, we have

From (1), differentiating on the supposition that z is constant,

Hence, f'(y). (y) taking the place of (y) in (3), we have

Proceeding in the same way it is evident that we shall have,

But from (2) and (5) it appears that yz, and

x = 0.