indefinite straight line UV drawn through P, at right angles to HK, the limiting position of H'K', that is, at right angles to the tangent at P, is called a normal to the curve at the point P.

Inclinations of the Tangent and the Normal at any point of a Curve to the coordinate Axes.

100. Let Ox, Oy, be rectangular axes of coordinates, (fig. 1), AB being a curve contained in the plane xOy. Let x, y, be the coordinates OM, MP, of P, and x', y', the coordinates of Q. Let e denote the length of the chord PQ. Then the cosine, sine, and tangent of the angle K' T'x, T'' being the point in which H'K' cuts Ox, are respectively

When approaches indefinitely near to x, in consequence of the approach of Q towards P, then ultimately, a denoting the angle KTx, where T is the point in which HK cuts Ox, s representing the arc AP, and s' the arc AQ,

But, by Newton's Seventh Lemma, in the first section of the Principia, we know that

Again, ẞ denoting the angle PGO, G being the point in which the normal UV cuts the axis of x,

We may express cos a and sin a in terms of dx and dy alone, without ds: thus, adding together the squares of the two

Again, supposing the equation to the curve to be

Ex. To find the inclination of the tangent at any point of an ellipse

to the coordinate axes.

y2

1

M

The negative sign in the expression for tan a, supposing r and y to be positive, shews that the angle PTx is obtuse, instead of acute, as in fig. (1).

Equations to the Tangent and the Normal at any Point of

a Curve.

101. Let R, fig. (2), be any point whatever in the tangent HK at the point P of the curve AB.

Let (x, y), (x', y'), be the coordinates of P, R, respectively.

Then, denoting the distance between P and R,

a form of the equation to the tangent at P. An equation, usually more convenient, may be obtained in the following

manner.

Differentiating the equation u = 0 to the curve, we have

multiplying the former and the latter term of this equation by the two equal quantities

respectively, we obtain, for the equation to the tangent,

The equation to a line through the point x, y, at right angles to the tangent, that is, the equation to the normal at P, will therefore be, as we know by the condition of perpendicularity given in treatises on algebraic geometry,

Ex. To find the equations to the tangent and normal at any

or, by virtue of the equation to the ellipse,

Distance of the Origin of Coordinates from the Tangent.

102. Let & denote the distance of the origin from the tangent, ε being the inclination of 8 to the axis of x. Then the equation to the tangent must coincide with the equation