28. If O be the pole and P any point of the curve

r = a cos me,

and if with O for pole and P for vertex a similar curve be described, the envelope of all such curves is

29. If O be the pole and P any point of the curve

and if with O for pole and P for vertex a curve similar to r2 = an cos no

be described, the envelope of all such curves is

30. If O be the pole and Y the foot of the perpendicular from O on any tangent to the curve

pm = am cos mb,

and if with O for pole and Y for vertex a curve similar to yn=an cos no

be described, the envelope of all such curves is

31. If a point on the circumference of a given circle be taken as pole, and circles be described on radii vectores of the given circle as diameters, the envelope of these circles is a cardioide.

32. Show that the envelope of all cardioides on radii vectores of the circle r = a cos 0 for axes, and having their cusps at the pole, is r2 = a3 cos 10.

33. Show that the envelope of all cardioides described on radii vectores of the cardioide r=a(1+ cos 0) for axes, and having their cusps at the pole, is

rt = (2a)1 cos

4

34. On radii vectores of p2n = a2n cos 2n0 as axes curves similar to it are described, the curves being all concentric. Show that the envelope of all these is

is

is

p2 = a2 cos n0.

35. Prove that the pedal equation of the envelope of the line x cos 20+ y sin 20 = 2a cos 0

p2 = 1 (r2 — a2).

36. Prove that the pedal equation of the envelope of the line x cos m✪ + y sin mə a cos n

по

m2,2 = (m2 — n2)p2 + n2a2.

37. Two central radii vectores of a circle of radius a rotate from coincidence in a given initial position with uniform angular velocities w and w'. Show that the pedal equation of the envelope of a line joining their extremities is

(w + w')2r2 = 4ww'p2 + (w− w')2a2.

38. The envelope of polars with respect to the circle

x2 + y2 = 2ax

of points which lie on the circle

is

x2 + y2 = 2bx

{(a − b)x+ab}2 = b2 {(x − a)2 + y2}.

39. A square slides with two of its adjacent sides passing through fixed points. Show that its remaining sides touch a pair of fixed circles, one diagonal passes through a fixed point, and that the envelope of the other is a circle.

40. An equilateral triangle moves so that two of its sides pass through two fixed points. Prove that the envelope of the third side is a circle.

41. Prove that the envelope of the circles obtained by varying the arbitrary parameter a in the equation

c2(ya)2+(cx- a2)2= (a2 + c2)2 consists of a straight line and a circle.

42. Two points are taken on an ellipse on the same side of the major axis and such that the sum of their abscissae is equal to the semi-major axis. Show that the line joining them envelopes a parabola which goes through the extremities of the minor axis and whose latus rectum is equal to that of the ellipse.

43. Given the centre and directrices of an ellipse, show that the envelope of the normals at the ends of the latera recta is 27y1±256cx3 = 0.

44. Prove that the envelope of a circle which passes through a fixed point F and subtends a constant angle at another fixed point F" is a limaçon.

45. Find the envelope of a parabola of which the directrix and one point are given.

46. Find the condition between a and b that the envelope of

47. S is a fixed point, and with any point P of a curve for centre and with radius PS + k a circle is described. Show that the envelopes for different values of k consist of two sets of parallel curves, one set being circles; and find what the original curve must be that both sets may be circles.

48. Rays emanate from a luminous point O and are reflected at a plane curve. OY is the perpendicular from O on the tangent at any point P, and OY is produced to a point Q, such that YQ=OY. Show that the caustic curve is the evolute of the locus of Q. Show that the caustic curve may also be regarded as the evolute of the envelope of a circle whose centre is P and radius OP.

[If a ray of light in the plane of a given bright curve be incident upon the curve the reflected ray and the incident ray make equal angles with

the normal to the curve at the point of incidence, and the reflected ray lies in the plane of the curve. If a given system of rays be incident upon the curve, the envelope of the reflected rays is called the caustic by reflection.]

α

49. Parallel rays are incident on a bright semicircular wire (radius a) and in its plane. Show that the caustic curve is the epicycloid formed by a point attached to a circle of radius 4 rolling upon the circumference of a circle of radius 2

a

50. Rays emanate from a point on the circumference of a reflecting circular arc. Show that the caustic after reflection is a cardioide.

51. Show that if rays emanate from the pole of an equiangular spiral and are reflected by the curve the caustic is a similar equiangular spiral.

CHAPTER XII.

CURVE TRACING.

315. Nature of the Problem. Cartesian Equations. If, in the Cartesian equation of any algebraic curve, various values of x be assigned, we obtain a number of equations whose roots give the corresponding values of the ordinates. The real roots of these equations can always be either found exactly or approximated closely to by methods explained in the Theory of Equations. We can by this means, laborious though it will in most cases be, find as many points as we like which satisfy the given equation of the curve; and by joining these points by a curved line drawn freely through them we can form a fairly good idea as to its shape. The experience, however, which we have gained in previous chapters will in general obviate any necessity of resort to the usually tedious process of approximating to the roots of equations of high degree; and we propose to give a list of suggestions for guidance in curve tracing which in most cases will enable us to form, without much difficulty, a sufficiently exact notion of the character of the curve represented by any specified equation.