39. Show that the curve

x5+y5=5ax2y

has two cusps of the first species at the origin, and that x + y = a

is an asymptote.

40. Show that a cubic curve cannot have more than one double point, and cannot have a triple point.

Examine the case of the curve

2(x3 + y3) − 3(3x2 + y2) + 12x = 4

and show that there are apparently two nodes at (1, 1) and at (2, 0) respectively. Explain this result.

41. Show that the curve

by2 = x3 sin22

α

has a cusp of the first species at the origin and is symmetrical with regard to the axis of x. Show also that it has an infinite series of conjugate points lying at equal distances from each other along the negative portion of the axis of x.

42. Show that the curve

y = x = log ey2

has a node at the point (1, 2).

43. Show that the curve

4x

(x2 + y2)2 = a(3x2y — y3)

has a triple point at the origin, and that the angles between the branches through the origin are equal.

has a multiple point of the eighth order at the origin, and that the curve consists of eight equal loops.

y2 =

the origin is a cusp of the first species.

46. Show that for the Conchoid

x2y2 = (a + y)2(b2 — y2),

if b be a there is a node at x= 0, y= a, and if ba there is

a cusp at the same point.

47. Show that at the point (-1, -2) there is a cusp of the first species on the curve

x2+2x2+2xy - y2+5x - 2y = 0.

48. Examine the singularities of the curve

x2 - 4ax3- 2ay3 + 4a2x2 + 3a2y2 – a1 = 0.

There are nodes at the points (0, a), (a, 0), (2a, a). Find the directions of the tangents at these points.

49. Show that the curve

x- 2x2y-xy2 - 2x2-2xy + y2-x+2y+1=0

has a single cusp of the second kind at the point (0, −1).

50. Examine the character of the curve

ay1 — ax2y2 + x + x1y = 0

in the immediate neighbourhood of the origin.

51. Show that at each of the four points of intersection of the curve

(ax)3 + (by)3 = (a2 — b2)

with the axes there is a cusp of the first species.

52. Show that the origin is a conjugate point on the curve x2 - ax2y + axy2 + a2y2 = 0.

53. Show that at the origin there is a single cusp of the second species on the curve x2 - 2ax2y - axy2 + a2y2 = 0.

54. Show that the curve

y2 = 2x2y + x3y + x3

has a single cusp of the first species at the origin. 55. Show that the curve

y2 = 2x2y + x1y + x*

has a double keratoid cusp at the origin.

56. Show that the curve

y2 = 2x2y + x1y - 2x1

has a conjugate point at the origin.

CHAPTER X.

CURVATURE.

CURVATURE.

265. Angle of Contingence.

Let PQ be an arc of a curve. Suppose that between P and Q there is no point of inflexion or other singularity, but that the bending is continuously in one direction. Let LPR and MQ be the tangents at P and Q, intersect

ing at T and cutting a given fixed straight line LZ in L and M. Then the angle RTQ is called the angle of contingence of the arc PQ.

The angle of contingence of any arc is therefore the

difference of the angles which the tangents at its extremities make with any given fixed straight line. It is also obviously the angle turned through by a line which rolls along the curve from one extremity of the arc to the other.

266. Measure of Curvature.

It is clear that the whole bending or curvature which the curve undergoes between P and Q is greater or less according as the angle of contingence RTQ is greater or angle of contingence less. The fraction is called the length of arc

average bending or average curvature of the arc. We shall define the curvature of a curve in the immediate neighbourhood of a given point to be the rate of deflection from the tangent at that point. And we shall take as a measure of this rate of deflection at the given point the angle of contingence

limit of the expression

length of arc

when the

length of the arc measured from the given point and therefore also the angle of contingence are indefinitely diminished.

That this is a proper measure of the rate of deflection is obvious from the consideration that, for a given length of arc, the deflection is greater or less as the angle of contingence is greater or less, and for a given angle of contingence the deflection is greater or less as the length of the arc is less or greater.

267. Curvature of a Circle.

In the case of the circle the curvature is the same at

every point and is measured by the RECIPROCAL OF THE

For let r be the radius, O the centre. Then

the angle being supposed measured in circular measure. angle of contingence__ 1

Hence

and this is true whether the limit be taken or not. Hence the "curvature" of a circle at any point is measured by the reciprocal of the radius.

268. Circle of Curvature.

If three contiguous points be taken on a curve, a circle may be drawn to pass through those three points. Let them be P, Q, R. Then, when the points are indefinitely close together, PQ and QR are ultimately tangents both to the curve and to the circle. Hence at the point of ultimate coincidence the curve and the circle have the same angle of contingence, viz., the angle RQZ (see Fig. 52). Moreover, the arcs PR of the circle and the curve differ by a small quantity of order higher than their own, and therefore may be considered equal in the limit (see Art. 36). Hence the curvatures of this circle and of the