CHAPTER VII.

TANGENTS AND NORMALS.

169. Equation of TANGENT.

It was shown in Art. 38 that the equation of the tangent at the point (x, y) on the curve y=f(x) is

X and Y being the current co-ordinates of any point on

the tangent.

Suppose the equation of the curve to be given in the

170. Simplification for Algebraic Curves.

If f(x, y) be an algebraic function of x and y of degree n, suppose it made homogeneous in x, y, and z by the introduction of a proper power of the linear unit z wherever necessary. Call the function thus altered f(x, y, z). Then f(x, y, z) is a homogeneous algebraic function of the nth degree; hence we have by Euler's Theorem af af af

(Art. 161)

x +y

+ y + z z2 = nf(x, y, z) = (), дх ду ez

by virtue of the equation to the curve.

Adding this to equation (2), the equation of the tangent takes the form

The equation, when made homogeneous in x, y, z by the introduction of a proper power of z, is

f(x, y, z)=x+a2xyz2+b3yz3+c+4=0,

Substituting these in Equation 3, and putting 2= =1, we have for the equation of the tangent to the curve at the point (x, y)

X(4x3+a2y)+Y(a2x+b3)+2a2xy+3b3y+4c1=0.

With very little practice the introduction of the z can

be performed mentally. It is generally more advantageous to use equation (3) than equation (2), because (3) gives the result in its simplest form, whereas if (2) be used it is often necessary to reduce by substitutions from the equation of the curve.

171. Application to General Rational Algebraic Curve. If the equation of the curve be written in the form

f(x, y) = Un+Un-1+Un-2+...+u2+u1+u=0 (where u, represents the sum of all the terms of the th degree), then when made homogeneous by the introduction where necessary of a proper power of z we shall have

f(x, y, z)=un+Un-1%+Un-222+...

and

Əz

=Un-1+2Un-22+3Un-322+.....

+(n−2)u ̧zn-3+(n−1)u ̧22-2+nu zn−1,

and therefore substituting in (3) and putting z=1, the equation of the tangent is

+(n−2)u2+(n−1)u1+nu=0............... (4)

172. NORMAL.

DEF. The normal at any point of a curve is a straight line through that point and perpendicular to the tangent to the curve at that point.

Let the axes be assumed rectangular. The equation of the normal may then be at once written down. For if the equation of the curve be

y = f(x),

L

If the equation of the curve be given in the form

This requires 22 in the last term to make a homogeneous equation in x, y, and z. We have then

Ex. 2. Take the general equation of a conic

ax2+2hxy+by2+2gx+2fy+c=0.

When made homogeneous this becomes

ax2+2hxy+by2+2gxz+2fyz+cz2=0.

The equation of the tangent is therefore