MECHANICS 685 portional to the angular velocity of the tangent (whose direction is parallel to the acceleration). rcular Thus, in the hodograph, the angle between successive lo- tangents is proportional to the arc between their points P h> of contact ; and therefore the curvature is constant ; i.e., the hodograph is a circle. rm of Let A (fig. 13) be the centre of this circle, O the pole of -ne- the hodograph, P any position of the tracing point. Then OP is, in magnitude and direction, n the velocity in the orbit. But it may be looked on as consisting of two parts, OA and AP. Of these both are constant in magnitude ; but OA is constant in direction, while AP is perpendicular to the direction of ac celeration in the orbit. Hence the velocity in the orbit is the resultant of two constant parts, one always in a fixed direction, the other always perpendicular to the radius-vector. This gives the form of the orbit as follows : x = a(e-y/r), y = ax/r ; so that rr = xx + yy = cry , or r = e(y + b); where the meanings of the quantities are obvious. But if PO cut the circle again in p, Op is proportional to the perpendicular on the tangent to the orbit from the centre of acceleration (because PO.Op is constant) and is at right angles to it (because it is in the direction of the velocity). Hence the path is such that the locus of the foot of the perpendicular from the centre of acceleration on the tangent is a circle. This property belongs exclu sively to conic sections, one focus being the point from which the perpendiculars are drawn. A third and even simpler mode of treating this most important problem is as follows. Draw OM perpendicular to PA (produced if necessary) and PN perpendicular to OA. Then OM is the resolved part of OP parallel to the tangent at P, i.e., it is the speed with which the length of the radius-vector changes. Also PN is the resolved part of OP perpendicular to the fixed line OA, i.e., it is the speed with which the moving point travels in a fixed direction. But by similar triangles 0AM, PAN, we have OM : PN : : OA : AP = a constant ratio. Hence the increment of the radius-vector bears a constant ratio to the simultaneous increment of the distance of the moving point from a fixed line in the plane of motion. This is only a slightly altered form of statement of the focus and directrix property of conic sections. When O is within the circle, the constant ratio is less than unity, and the conic is an ellipse ; when without, the ratio is greater than unity, and we have an hyperbola. When O is on the circumference of the hodograph, the path is a parabola ; for the ratio is unity. In a subsequent section we will return to this question, and treat it from the point of view of Kepler s Laws of Planetary Motion. .naly- Simple as are the geometrical methods above, the direct analy-
- ical tical one is still simpler. For we have
proof. so that x = ^- cos 6 . j)= ^- sin . r- r- Hence, as before ( 47), xy-yx r^^h ; and therefore, by eliminating r z , we have so that These give at once, by squaring and adding, (x-)2 + (y-0) 2 = M 2 /^, the equation of the circular hodograph. Also, by multiplying the first by y, and subtracting it from the second multiplied by x, we have h - |3x + ay = rpjh , the equation of the orbit. This is evidently a conic section of which the origin is a focus. The directrix corresponding is the line ay - fix + h = , and the excentricity is h Va- + 0-//* From these the major axis can be calculated. 50. Elliptic Motion about the Centre. When a point Central moves uniformly in a circle, the motion presents very differ- accelera- ent appearances according to the spectator s point of view. tlon P ro ~ If we suppose him to be situated at a distance very great to ra- compared with the radius of the circle, he sees what is prac- dius- tically an orthographic projection of the orbit on a plane vector. perpendicular to the line of sight. In general, an ortho graphic projection of a circle is an ellipse whose centre is the projection of that of the circle. As equal areas are projected orthographically into equal areas, the appearance is therefore elliptic motion, in which the radius-vector from the centre describes equal areas in equal times. Hence ( 46) the acceleration is directed towards the centre. But accelerations are projected like velocities, and like lines. Hence, as the acceleration in uniform circular motion is constant, and directed towards the centre, so in elliptic motion, with equable description of areas about the centre, . the acceleration is towards the centre, and is proportional to the length of the radius-vector}- But this projected orbit may again be projected orthographically, as often as we please, on different planes. It will always remain elliptical, and with the radius-vector from the centre describing equal areas in equal times. And the acceleration will always be in the same proportion as before to the radius-vector. However different in size and shape these elliptic orbits may be, they have one common property, the time of describing them is the same. Thus we see that when the orbit is an ellipse described about its centre of figure the acceleration is central, and proportional to the radius-vector. The time of describing such an ellipse depends only upon the ratio of the acceleration to the length of the radius-vector ; or, if we choose, upon the magnitude of the acceleration at unit distance. And the converse of this proposition is also evidently true. When we look edgewise at the uniformly- described circular path with which we commenced, it is seen projected into a straight line, in which the moving point appears to oscillate. This is the case, for instance, very approximately, with the satellites of Jupiter as seen from the earth. Sun-spots, the red-spot on Jupiter, &c., all appear to move approximately in this way. But the extreme importance of this species of motion is that it is the simplest type of oscillation of a particle of matter displaced from a position of stable equilibrium. The vibrations of the ether when homogeneous plane-polarized light is passing through it, of the air when a pure musical note is sounded, the oscillations of a pendulum (through small arcs), the simplest vibrations of a pianoforte wire or a tuning-fork, the indications of a tide-gauge when the sea is calm, all are instances of it. Hence the special necessity for studying it in detail. 51. DBF. Simple harmonic motion is the resolved part, Simple parallel to a diameter, of uniform circular motion. x-a= --sine, y-fi = 1 The elastic force called into play by dispLicement is, by Ilooke s law, proportional to the displacement, and tends to restore the dis placed particle to its equilibrium position. We mention this, in passing, to show the importance of the present investigation. harmo nic mo
tion.