then the accelerations parallel to the axes are, on this scale, the velocities of P, and are therefore Or, the acceleration of P relative to N, in the direction Ox, being i-ves, and the acceleration of N in the same direction being we2, the acceleration of P parallel to Ox is the sum of these two, and the accelerations parallel to Oy and Oz are obtained in the same manner. EXAMPLES. 1. Assuming that the earth describes a circle uniformly about the sun in a year, that the distance of their centres is 240 radii of the sun, and that the radius of the sun is 100 times that of the earth, find the measure of the velocity of the vertex of the earth's shadow, taking the sun's radius as the unit of length and a year as the unit of time. 2. If one point move uniformly in a circle, and another move with equal velocity in a tangent to the circle, what are their relative paths? 3. The radius of the earth being 4000 miles, the latitude, λ, of a place at which a train travelling westward at the rate of 1 mile per minute is at rest in space is given by 4. A particle B describes a circle uniformly about the fixed point A, and C describes a circle uniformly in the same plane about B. Find the motion of C relative to A. 5. A circle revolves with uniform velocity about its centre. The centre moves with varying velocity along a straight line. Find the velocity parallel to this line at any instant of a point on the circumference, and deduce the acceleration of the centre necessary for this point to be always moving at right angles to the line. 6. A point moves in a curve in such a way that its direction of motion changes at a rate varying as the velocity directly and the whole space described inversely. Prove that the curvature varies inversely as the arc. 7. A wheel revolves uniformly about its centre C, which is fixed, and a particle A moves uniformly in a straight line through the centre; describe the path of a point B in the wheel relative to A, (1) when CA is in the plane of the wheel, (2) when CA is perpendicular to that plane. 8. If the resolved parts of the velocity of a moving particle perpendicular to its distances from two fixed points are constant, and equal to one another, its velocity varies as the square root of the product of its distances from these points. 9. If the acceleration of a falling body be the unit of acceleration and a velocity of 60 miles an hour the unit of velocity, find the units of length and time. 10. In two different systems of units an acceleration is represented by the same number, while a velocity is represented by numbers in the ratio 1: 3. Compare the units of time and space. 11. Prove that the locus of the points about which the angular velocity of a point moving in any manner is, at the same instant, the same, is a circle. 12. If the angular velocity of a particle about a given point in its plane of motion be constant, prove that the transversal component of its acceleration is proportional to the radial component of its velocity. 13. If the velocities of a point parallel and perpendicular to the radius vector are always proportional to each other, and likewise the accelerations, its velocity will vary as some power of the radius vector. 14. If the velocity of a point be resolved into any number of components in a plane, its angular velocity about any fixed point in the plane is the sum of the angular velocities due to the several components. 15. If the angular velocity @ of a particle about the origin is constant, and the rate of change of acceleration' is directed wholly along the radius vector, prove that der 16. A point P moves with uniform velocity in a circle; Q is a point in the same radius at double the distance from the centre, PR is a tangent at P equal to the arc described by P from the beginning of the motion; shew that the acceleration of the point R is represented in magnitude and direction by RQ. 17. The tangent at a point P of a parabola meets the tangent at the vertex in Y and the axis in T. If Y move with uniform velocity, shew that T moves with uniform acceleration if T move with uniform velocity, the velocity of Y varies inversely as AY. : 18. If the time is a quadratic function of the length of the path of a moving point, prove that the harmonic mean of the initial and final velocities is equal to the velocity at the middle point of the path, and that the tangential retardation is proportional to the cube of the velocity. 19. Shew that it is possible for a point to move so that the velocity at any time shall be proportional to the space described from a fixed origin at a time a seconds before, and the acceleration at any time shall be proportional to the velocity a seconds after, and determine the law of the motion. 20. A point A moves in a straight line, and a second point B always moves towards A and keeps at a constant distance from it. Find the path of B and shew that its velocity is a mean proportional between the velocity of its projection on the path of A and the velocity of A. 21. A point moves in an ellipse so that the velocity varies as the square of the diameter parallel to the direction of motion: prove that the resultant acceleration at any instant will be in the direction of the line joining the point with the middle point of the perpendicular from the centre on the tangent at the point. 22. A point moves in a plane in such a manner that its tangential and normal accelerations are always equal, and its velocity varies as etan-1, s being the length of the arc of the curve measured from a fixed point; find the path. 23. If a curve roll in contact with a straight line with uniform velocity, shew that the acceleration of the point in contact with the straight line varies inversely as p, but if with uniform angular velocity directly as p; p being the radius of curvature of the curve at the point of contact. 24. A curve rolls along a straight line, the point of contact moving uniformly along the line. Shew that the acceleration of the centre of curvature of the rolling curve at the point of contact is, at the instant, proportional to 1 dp p ds 25. Prove that the angular acceleration of the direction of motion of a point moving in a plane is 26. The position of a point is given by the perpendiculars , n on two fixed lines making an angle a with each other, prove that the component velocities in the directions of §, n are respectively (§ + ŋ cos a)/sin2 a and (ʼn + § cos a)/sin2 a. 27. A point moves with constant linear velocity wa, and its angular velocity about the pole is wr/a; shew that its path is a lemniscate or a circle and explain how these solutions are related. Shew further that its acceleration is equal to 3w2r. 28. If the axes Ox and Oy revolve with uniform angular velocity w, and the component velocities of the point (x, y) parallel to the axes be A/x and B/y, then the square of the distance of the point from the origin will increase uniformly with the time. 29. A point moves in a plane curve and sounds as it moves. At a fixed point C in the plane the whole sound produced is heard simultaneously. Shew (i) that if the point moves uniformly, the curve is an equiangular spiral—(ii) if the velocity of the point vary inversely as the distance of C from its line of motion, the curve is a reciprocal spiral. 30. A point moves in the arc of a cycloid so that the tangent turns uniformly; prove that the acceleration of the point is constant. 31. If the axes Ox, Oy revolve with constant angular velocity w, and the component velocities of the point (xy) parallel to the axes are prove that the point describes relatively to the axes an ellipse in the periodic time 32. If the motion be referred to two axes one of which is fixed, and the other revolves about the origin in such a way that the line joining the origin to the particle is equally inclined at an angle to the axes, shew that the component acceleration parallel to the fixed axis (§) is § — (2έ0 + §¤) cosec 0. What is the other component ? 33. If the radial and transversal accelerations of a particle be each proportional to the velocity in the direction of the other, the path of the particle is given by an equation of the form |