CHAPTER IX.

ON THE TANGENTS, NORMALS AND ASYMPTOTES TO CURVES.

SECT. 1. Rectilinear Co-ordinates.

Ir the equation to the curve be put under the form
y = f(x),

the equation to a tangent at a point y is

and y' being the current co-ordinates of the tangent.

If the equation to the curve be put under the form
u = p(x, y) = c,

the equation to the tangent takes the more symmetrical form

If u be a homogeneous function of n dimensions in and y, by a well-known property of such functions

if u be a homogeneous function of ʼn dimensions in x and y. The portion of the tangent intercepted between the point of contact and the perpendicular on it from the origin is

The portions of the axes cut off between the origin and the tangent, or the intercepts of the tangent, are

These I shall call yo, a respectively.

Ex. (1). The equation to the hyperbola referred to its asymptotes is

Hence the product of the intercepts of the tangent

= x。 Y = 4xy = 4m2 is constant ;

and the triangle contained between the axes and the tangent, being proportional to this product, is also constant.

(2) The equation to the parabola referred to two tangents

or a, yo are the co-ordinates of the chord joining the points at which the axes touch the curve.

(3)

The equation to one of the hypocycloids referred

to rectangular co-ordinates is

=

Therefore, ax3, y = a3y; and the portion of the tangent intercepted between the axes =

(x2 + y2)3 = a; or the hypocycloid is constantly touched by a straight line of given

length which slides between two rectangular axes. The converse of this proposition, viz. that the locus of the ultimate intersections of a line of given length sliding between rectangular axes is this hypocycloid, was first shewn by John Bernoulli. (See his Works, Vol. 111. p. 447.)

For the perpendicular from the origin on the tangent we find

The subtangent = a, and is therefore constant.
The tangent = (a2 + y2)3.

y

The subnormal. The normal = - 1⁄2 (a2 + y2)3.

a

(9) The equation to the cycloid referred to its vertex is

If M be the point where the ordinate meets the generating circle, and if we join MA, MB, then

That is to say, the tangent to the cycloid is parallel to the chord of the generating circle. The normal is evidently parallel to the other chord MB. Hence also the angle which two tangents make with each other is equal to the angle between the corresponding chords of the generating circle.

Y。=y-(2ax - x)= PN- MN = PM.

But from the generation of the curve, PM is equal to the arc of the circle AM, therefore yo = arc AM.