The value of R,' is equal to the definite integral 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, ẞ). Putting a + x, ẞ+y, for x, y, we have, for the transformed equation, 0 = f(a + x, ß + y), or, by Taylor's theorem, the dimensions of the proposed equa tion in x and y being n, then ... + y" 0 = Ax2 + By3 + 2Cxy + 2A'x + 2By + C' : 0 = Aa2 + Bß2 + 2 Caß + 2A'a + 2B'ß + C' dB 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 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, 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, ƒ'(y). $(y) taking the place of ¥(y) in (3), we have Proceeding in the same way it is evident that we shall have, generally, d"u = dn-1 dzi [ƒ'(y). {$(y)}" . dy]. By Stirling's theorem, . But from (2) and (5) it appears that yz, and dy dz x = 0. + or, when f(y) = y, we have ƒ'(z) = 1, and, instead of the formula (7), we have This theorem was given by Lagrange, in the Mémoires de Berlin for the year 1770, (see also his Equations Numeriques, Note XI.), as a generalization of a a generalization of a particular development obtained by Lambert for the expression of the roots of certain algebraic equations: Lambert's results were published in the year 1758. A demonstration of Lagrange's theorem, due to Cauchy, which involves some important reflections respecting the convergency of the series, may be seen in Moigno's Leçons de Calcul Différentiel et de Calcul Intégral, tom. 1., pp. 162–172. An expression for the limits of error committed in stopping at any term of the series, has been given by Murphy in the fourth volume of the Cambridge Philosophical Transactions. The determination of u in terms of x and z is a celebrated problem in astronomy called Kepler's problem: the variable z denotes a quantity which varies as the time, the coefficient x represents the eccentricity of the elliptical orbit of a planet, and the variable u is an angle called the Eccentric Anomaly. (z) = sin 2, |