Thus let the centre of the ellipse be origin; = y, OQ = {, which equation may be dealt with in the same manner as that to the cyclonoid, and the curve traced in a similar fashion. an We may notice as a curious illustration of the contrasts afforded by the variety of nature when classified under general principles; that this curve is alike that which is described by a flower in the hem of a lady's 8. dress as she waltzes round oval saloon; and the course which the moon fulfils as she perpetually encircles the elliptic orbit of the earth. So that we may call this locus, the lunar or waltz curve (8.) Thus we have a new class of lines forming festoons around those which are more familiar and of which the equations are more easily analysed. The characteristic of this class is equation(a) viz., s = cp where s is an arc of the locus, and and c, respectively, the angle and radius of revolution. The equations (y) and (d) which remain unchanged for all festoons, due regard being paid to the origin chosen for the co-ordinates and the starting point of the locus, will enable the student to trace the festoon in relation to its directing curve. And if ,, and 4 be eliminated from these three equations, G combined with that of the particular directing curve, we shall have the equation to the locus. But it will in general be transcendental and involve a function of s, which, however, may in some cases be eliminated by means of differentiation, and the substitution of dx2+ dy2 for ds2; and the integration of this differential equation will produce one involving a function of x and y only. In this class of curves the Cycloid is the Festoon of a straight line (except that the circumference and not the centre of the Generating circle moves on the Directing straight line,) the Cyclonoid is the Festoon of a Parabola, and the Waltz curve of the Ellipse. The Hyperbola, the Catenary, and others, will all have Festoons, as we have just observed. Such are a few, and these the simplest, of the infinite number of curves wherewith nature abounds. They form the chief portion of those which have yielded to analysis, but they are far, indeed, from exhausting the almost inexhaustible subject. "All nature is in motion," and every motion has its own peculiar curve. The curling waves, the leaves of trees, the varied flowers and fruits, the minaret and arch of Goth or Turk, the exquisite and changeful outline of animal form, nay, the perfection of the human figure itself, all depend for their elegance upon the suppleness and grace of their delineating curves. So that in fact it may be justly said that delicacy of outline is to the painter what accuracy of time is to the musician, the secret difficulty but chief beauty of his glorious art. ONE property we must not forget, which is common to all the curves we have been considering, viz.,-Contact. We have already seen that the circle and straight line may have two points in common, and that these may be consecutive. And, in general, any two curves, one of which can be made to pass through a greater number of points than the other, may have as many points in common with it as the latter can be made to pass through. And all these points may be consecutive. Thus a circle may be made to pass through three points of an ellipse. And when these three points are consecutive, the circle is called the Circle of Curvature, and measures the eccentricity of the ellipse, that is, the amount of its deviation from a circle, as a tangent measures its deflection from a straight line. MEANS OF DETERMINING SUCCESSIVE POSITIONS OF A POINT WHEN THE LAW OF MOTION IS GIVEN. (Differential Calculus.) In the case of the circle and straight line it was easy to discover whether a line was one or the other by trying whether the angle between any three consecutive points contained 180 degrees or not. But such a method is plainly inapplicable to curves in general, inasmuch as the number of points which two curves in general may have in common varies with the kind of curve. At the same time the form of the curve must evidently be definite if the motion from every point in the curve to the succeeding point be known. Some device, therefore, becomes necessary by which we may be enabled to express not merely the relation which exists between the co-ordinates in general, but also that between every two successive positions of a point moving in any given curve. It is afforded by the application of the problem, the results of which we are still discussing, to the method of Rectangular Co-ordinates. For let P and Q be two consecutive positions of a point moving in any curve whatsoever. Then the portion of the curve PQ may be considered a straight line. (Chap. xv.) Let O be the origin; P R M N OM and MP be the co-ordinates of P; ON and NQ be the co-ordinates of Q; and let PR be drawn parallel to ON, then MN equals PR, and PQ2=PR2+RQ2 And thus, PQ is definite as well as PR, and RQ. And since Q is the next point in the curve to P, the increase of its co-ordinates is due only to the motion of the point. That is to say, PR (or MN) and RQ are the amount by which the co-ordinates of any point are increased by the motion of the point from any one position in the curve to the next. And we have seen in the preceding chapter that the relation between the co-ordinates of the points in any given curve remains unchanged, and may be expressed by means of an equation. If, then, the equation which expresses this relation be given for any particular curve-say for an ellipse, we are enabled, by introducing the modifications due to the fact that PQ, PR, RQ are all infinitesimal to calculate the relations which exist between them. In other words, given the law of motion, we are enabled to calculate the increase or rather the "increments" of the co-ordinates due to the motion of the point, from any one position in the curve to the next. Thus we obtain two equations—one which expresses the relation of the co-ordinates of any one point in the curve to those of any other; the other expressing the relation of the co-ordinates of any one point in the curve to those of the succeeding point, this latter equation being in truth the former, modified by the fact that the relation it expresses is between infinitesimals. Hence it is "derived" from the former by means of the rules of the Differential Calculus, and is called the "Derived equation to the curve." Given then the equation to a curve, it is possible by the rules just mentioned to find the "derived equation." And similarly, if the derived equation to a curve were known, and means for reversing the processes of the Differential Calculus were obtained, it would be possible from the derived equation to deduce the equation for the whole motion. This problem will be found in MEANS OF ASCERTAINING THE GENERAL LAW OF MOTION WHEN THAT FOR ANY TWO SUCCESSIVE POSITIONS OF THE POINT IS GIVEN. (Integral Calculus.) LASTLY, let us consider the case of a body moving in an orbit, the law of which is unknown to us, and the equation of which therefore we have no means of forming. Let us conceive, how |