the case, for any position ; the determination of the form of the path described, and of other circumstances of the motion, is a question of pure mathematics, and in all cases is capable (if not of an exact solution, at all events) of a solution to any degree of approximation that may be desired. This is true also if the total acceleration and its direction at every instant, or simply its rectangular components, be given, provided the velocity and its direction, as well as the position of the point, at any one instant be given. But these are in general questions requiring for their solution a knowledge of the integral calculus. 41. From the principles already laid down, a great many interesting results may be deduced, of which we enunciate a few of the simpler and more important. (a) If the velocity of a moving point be uniform, and if its direction revolve uniformly in a plane, the path described is a circle. (6) If a point moves in a plane, and its component velocity parallel to each of two rectangular axes is proportional to its distance from that axis, the path is an ellipse or hyperbola whose principal diameters coincide with those axes; and the acceleration is directed to or from the centre of the curve at every instant ($$ 66, 78). (c) If the components of the velocity parallel to each axis be equimultiples of the distances from the other axis, the path is a straight line passing through the origin. (d) When the velocity is uniform, but in a direction revolving uniformly in a right circular cone, the motion of the point is in a circular helix whose axis is parallel to that of the cone. 42. When a point moves uniformly in a circle of radius R, with velocity V, the whole acceleration is directed towards the centre, and V2 has the constant value See $ 36. R 43. With uniform acceleration in the direction of motion, a point describes spaces proportional to the squares of the times elapsed since the commencement of the motion. This is the case of a body falling vertically in vacuo under the action of gravity. In this case the space described in any interval is that which would be described in the same time by a point moving uniformly with a velocity equal to that at the middle of the interval. In other words, the average velocity (when the acceleration is uniform) is, during any interval, the arithmetical mean of the initial and final velocities. For, since the velocity increases uniformly, its value at any time before the middle of the interval is as much less than this mean as its value at the same time after the middle of the interval is greater than the mean: and hence its value at the middle of the interval must be the mean of its first and last values. In symbols; if at time t=0 the velocity was V, then at time t it is v=V+at. Also the space (x) described is equal to the product of the time by the or average velocity. But we have just shown that the average velocity is ={(V+ V + at)=V + lat, and therefore x=Vt + fata. Tv-V=ax. vrat, x=Lat", =jat?, ?v=ax. Of course the preceding formulae apply to a constant retardation, as in the case of a projectile moving vertically upwards, by simply giving a a negative sign. 44. When there is uniform acceleration in a constant direction, the path described is a parabola, whose axis is parallel to that direction. This is the case of a projectile moving in vacuo. For the velocity (V) in the original direction of motion remains unchanged; and therefore, in time t, a space Vt is described parallel to this line. But in the same interval, by the above reasoning, we see that a space fata is described parallel to the direction of acceleration. Hence, if AP be the direction of motion Р at A, AB the direction of acceleration, c and Q the position of the point at time t; AP = Vt, PQ = {ata. 2 V2 PO. the axis is parallel to AB; AB being a A diameter, and AP a tangent. If o be the focus of this curve, we know that AP=40A PQ Hence IB V2 2a' and is therefore known. Also OA is known in direction, for AP bisects the angle, OAC, between the focal distance of a point and the diameter through it. 45. When the acceleration is directed to a fixed point, the path is in a plane passing through that point; and in this plane the areas traced out by the radius-vector are proportional to the times employed. This includes the case of a satellite or planet revolving about its primary, according to Kepler's first law. Evidently there is no acceleration perpendicular to the plane containing the fixed point and the line of motion of the moving point at any instant; and there being no velocity perpendicular to this plane at starting, there is therefore none throughout the motion; AP? = a OA= thus the point moves in the plane. For the proof of the second part of the proposition we must make a slight digression. 46. The Moment of a velocity or of a force about any point is the product of its magnitude into the perpendicular from the point upon its direction. The moment of the resultant velocity of a particle about any point in the plane of the components is equal to the algebraic sum of the moments of the components, the proper sign of each moment depending on the direction of motion about the point. The same is true of moments of forces and of moments of momentum, as defined in Chapter II. First, consider two component motions, AB and AC, and let AD be their resultant ($ 31). Their half-moments round the point 0 are respectively the areas OAB, OCA. Now OCA, together with half the area of the parallelogram CABD, is equal to OBD. Hence the sum of the two half-moments together with half the area of the parallelogram is equal to AOB together with BOD, that is to say, to the area of the whole figure OABD. But ABD, a part of this figure, is equal to half the area of the parallelo C gram; and therefore the remainder, OAD, is equal to the sum of the two half-moments. But OAD is half the moment of the В resultant velocity round the point 0. Hence the moment of the resultant is equal to the sum of the moments of the two components. By attending to the signs of the moments, we see that the proposition holds when 0 is within the angle CAB. If there be any number of component rectilineal motions, we may compound them in order, any two taken together first, then a third, and so on; and it follows that the sum of their moments is equal to the moment of their resultant. It follows, of course, that the sum of the moments of any number of component velocities, all in one plane, into which the velocity of any point may be resolved, is equal to the moment of their resultant, round any point in their plane. It follows also, that if velocities, in different directions all in one plane, be successively given to a moving point, so that at any time its velocity is their resultant, the moment of its velocity at any time is the sum of the moments of all the velocities which have been successively given to it. 47. Thus if one of the components always passes through the point, its moment vanishes. This is the case of a motion in which the acceleration is directed to a fixed point, and we thus prove the second theorem of $ 45, that in the case supposed the areas described by the radius-vector are proportional to the times; for, as we have seen, the moment of the velocity is double the area traced out by the radius-vector in unit of time. 48. Hence in this case the velocity at any point is inversely as the perpendicular from the fixed point to the tangent to the path or the momentary direction of motion. For the product of this perpendicular and the velocity at any instant gives double the area described in one second about the fixed point, which has just been shown to be a constant quantity. Other examples of these principles will be met with in the chapters on Kinetics. 49. If, as in § 35, from any fixed point, lines be drawn at every instant representing in magnitude and direction the velocity of a point describing any path in any manner, the extremities of these lines form a curve which is called the Hodograph. The fixed point from which these lines are drawn is called the hodographic origin. The invention of this construction is due to Sir W. R. Hamilton; and one of the most beautiful of the many remarkable theorems to which it leads is this: The Hodograph for the motion of a planet or comet is always a circle, whatever be the form and dimensions of the orbit. The proof will be given immediately. It was shown ($ 35) that an arc of the hodograph represents the change of velocity of the moving point during the corresponding time; and also that the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph is equal to the amount of the acceleration of the moving point. When the hodograph and its origin, and the velocity along it, or the time corresponding to each point of it, are given, the orbit may easily be shown to be determinate. [An important improvement in nautical charts has been suggested by Archibald Smith? It consists in drawing a curve, which may be called the tidal hodograph with reference to any point of a chart for which the tidal currents are to be specified throughout the chief tidal period (twelve lunar hours). Numbers from I. to XII. are placed at marked points along the curve, corresponding to the lunar hours. Smith's curve is precisely the Hamiltonian hodograph for an imaginary particle moving at each instant with the same velocity and the same direction as the particle of fluid passing, at the same instant, through the point referred to.] 50. In the case of a projectile ($ 44), the horizontal velocity is unchanged, and the vertical velocity increases uniformly. Hence the hodograph is a vertical straight line, whose distance from the origin is the horizontal velocity, and which is described uniformly. 51. To prove Hamilton's proposition ($ 49), let APB be a portion of a conic section and S one focus. Let P move so that SP describes equal areas in equal times, that is ($ 48), let the velocity be inversely as the perpendicular -B SY from S to the tangent to the orbit. If ABP be an ellipse or hyperbola, the intersection of the perpendicular with the tangent lies in the circle S YAZ, whose diameter is the major axis. Produce YS to cut the circle again in Z. Then YS SZ is constant, and therefore SZ is inversely as SY, that is, SZ is proportional to the velocity at P. Also | Proc. R. S. 1865. U SZ is perpendicular to the direction of motion PY, and thus the circular locus of Z is the hodograph turned through a right angle about S in the plane of the orbit. If APB be a parabola, ÀY is a straight line. But if another point U be taken in YS produced, so that YS SU is constant, the locus of U is easily seen to be a circle. Hence the proposition is generally true for all conic sections. The hodograph surrounds its origin if the orbit be an ellipse, passes through it when the orbit is a parabola, and the origin is without the hodograph if the orbit is a hyperbola. 52. A reversal of the demonstration of § 51 shows that, if the acceleration be towards a fixed point, and if the hodograph be a circle, the orbit must be a conic section of which the fixed point is a focus. Bi we may also prove this important proposition as follows: Let A be the centre of the circle, and the hodographic origin. Join OA and draw the perpendiculars PM to OA and ON to PA. Then OP is the velocity in the orbit: and ON, being parallel to the tangent at P, is the direction of acceleration in the orbit; and is therefore parallel to the radius-vector to the fixed point about which there is equable description of areas. The velocity parallel to the radius-vector is therefore ON, and the velocity perpendicular to the fixed line ON ОА OA is PM. But PM=AP = constant. Hence, in the orbit, the velocity along the radius-vector is proportional to that perpendicular to a fixed line: and therefore the radius-vector of any point is proportional to the distance of that point from a fixed line a property belonging exclusively to the conic sections referred to their focus and directrix. 53. The path which, in consequence of Aberration, a fixed star seems to describe, is the hodograph of the earth's orbit, and is therefore a circle whose plane is parallel to the plane of the ecliptic. 54. When a point moves in any manner, the line joining it with a fixed point generally changes its direction. If, for simplicity, we consider the motion to be confined to a plane passing through the fixed point, the angle which the joining line makes with a fixed line in the plane is continually altering, and its rate of alteration at any instant is called the Angular Velocity of the first point about the second. If uniform, it is of course measured by the angle described in unit of time; if variable, by the angle which would have been described in unit of time if the angular velocity at the instant in question were maintained constant for so long. In this respect the |