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CHAPTER XII.

ON THE CURVATURE OF CURVED LINES.

SECT. 1. Radius of Curvature.

WHEN the curve is referred to rectangular co-ordinates, be the radius of curvature

the arc being made the independent variable.

If 0, a quantity of which both x and y are functional, be taken as the independent variable,

If u = 0 be the equation to the curve, the following expression for the radius of curvature is frequently convenient,

or, if u consist of the sum of two parts, the one involving alone and the other y alone,

(1) In the parabola, the equation to which is

totes

(3) In the rectangular hyperbola referred to its asymp

(4) In all the curves of the second order the radius of curvature varies as the cube of the normal.

If N be the length of the normal, N2 = 9o {1

y2 +

v

All the curves of the second order are included in the

(6) In the semi-cubical parabola 3ay2 = 2x3,

dy

(9) In the tractory y + (a2 - y2)}

= 0.

dx

Taking the expression for ρ in which y is the independent

(10) In the hypocycloid x + y = a3, p2 = 9 (a xy)3. If the curve be referred to polar co-ordinates r and 0, then

or, if it be expressed by the relation between r and the perpendicular on the tangent (p),

=

(12) In the lemniscate of Bernoulli a2 cos 20,

(13) In the spiral of Archimedes → = - αθ,

=

(15) The equation to the lituus being r2 Ꮎ

(16) The equation to the trisectrix being r=a (2 cos0±1),

(17) In the logarithmic spiral when referred to p and r,

(18) In the involute of the circle p2 = r2 — a2, and p = p.

—

(19) The equation to Cotes' spirals is p

When a curve is referred to rectanglar co-ordinates, the co-ordinates (a, ẞ) of its centre of curvature are given by the