d √ [{p (z)}'] = = {(sin x)'} = 2 sin z cos z = sin 2z, dz d dz [{ø Hence, as far as the term involving a3, we have, by formula (8), Hence, as far as the term involving x3, we have by formula (7), Differentiating (2), considering z constant, we have dy dy dx)' [1 - F" {z + xp (y)}. x p' (y)] = F" {z + x + (y)}.$(y)...(3); and, considering a constant, dx From (1), differentiating on the supposition that z is constant, Differentiating again, and bearing in mind the theorem established in the preceding article, viz. Proceeding in the same way we have, generally, From (2) and (4) it appears that, y = F(z), x = 0, and therefore we have, putting for the sake of brevity ƒ{F(2)} = ƒ,(z), and which is Laplace's formula, first given by him in the Mémoires de l'Académie des Sciences, 1777, p. 99. Lagrange's formula is evidently a particular case of Laplace's, from which it is at once derived by putting F(z) = 2. DIFFERENTIAL CALCULUS. SECOND PART. GEOMETRICAL APPLICATIONS. CHAPTER I. TANGENCY. Definition of a Tangent and of a Normal. 99. Let P, Q, be two points of a curve AB, (fig. 1), and suppose that an indefinite straight line H'K' is drawn through these two points. Conceive the point Q to move towards P; then the secant H'K' will keep tending towards a certain limiting position, and ultimately, that is, just as Q is on the point of coalescing with P, is said to be a tangent to the curve at P. An |