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