uniform or variable. It is said to be uniform when the point receives equal increments of velocity in equal times, and is then measured by the actual increase of velocity per unit of time. If we choose as the unit of acceleration that which adds a unit of velocity per unit of time to the velocity of a point, an acceleration measured by a will add a units of velocity in unit of time-and, therefore, at units of velocity in units of time. Hence if v be the change in the velocity during the interval,

v = at, or a = 2.

33. Acceleration is variable when the point's velocity does not receive equal increments in successive equal periods of time. It is then measured by the increment of velocity, which would have been generated in a unit of time had the acceleration remained throughout that unit the same as at its commencement. The average acceleration during any time is the whole velocity gained during that time, divided by the time. In Newton's notation & is used to express the acceleration in the direction of motion; and, if vs as in § 28, we have a=¿ = s.

34. But there is another form in which acceleration may manifest itself. Even if a point's velocity remain unchanged, yet if its direc tion of motion change, the resolved parts of its velocity in fixed directions will, in general, be accelerated.

Since acceleration is merely a change of the component velocity in a stated direction, it is evident that the laws of composition and resolution of accelerations are the same as those of velocities.

We therefore expand the definition just given, thus:-Acceleration is the rate of change of velocity whether that change take place in the 'direction of motion or not.

35. What is meant by change of velocity is evident from § 31. For if a velocity OA become OC, its change is AC, or OB.

Hence, just as the direction of motion of a point is the tangent to its path, so the direction of acceleration of a moving point is to be found by the following construction:

From any point O, draw lines OP, OQ, etc., repre senting in magnitude and direction the velocity of the moving point at every instant. (Compare § 49.) The points, P, Q, etc., must form a continuous curve, for (87) OP cannot change abruptly in direction. Now if Q be a point near to P, OP and OQ represent two successive values of the velocity. Hence PQ is the whole change of velocity during the interval. As the

interval becomes smaller, the direction PQ more and more nearly becomes the tangent at P. Hence the direction of acceleration is that of the tangent to the curve thus described.

The magnitude of the acceleration is the rate of change of velocity, and is therefore measured by the velocity of P in the curve PQ.

36. Let a point describe a circle, ABD, radius R, with uniform velocity V. Then, to determine the direction of acceleration, we must draw, as below, from a fixed point O, lines OP, OQ, etc., representing the velocity at A, B, etc., in direction and magnitude. Since the velocity in ABD is constant, all the lines OP, OQ, etc.,

B

The

D

will be equal (to V), and therefore PQS is a circle whose centre is O. The direction of acceleration at A is parallel to the tangent at P, that is, is perpendicular to OP, i.e. to Aa, and is therefore that of the radius AC.

Now P describes the circle PQS, while A describes ABD. Hence the velocity of P is to

that of A as OP to CA, i. e. as V to R; and is therefore equal to

and this (§ 35) is the amount of the acceleration in the circular path ABD.

87. The whole acceleration. in any direction is the sum of the components (in that direction) of the accelerations parallel to any three rectangular axes-each component acceleration being found by the same rule as component velocities (§ 34), that is, by multiply. ing by the cosine of the angle between the direction of the accelera tion and the line along which it is to be resolved.

38. When a point moves in a curve the whole acceleration may be resolved into two parts, one in the direction of the motion and equal to the acceleration of the velocity; the other towards the centre of curvature (perpendicular therefore to the direction of motion), whose magnitude is proportional to the square of the velocity and also to the curvature of the path. The former of these changes the velocity, the other affects only the form of the path, or the direction of motion. Hence if a moving point be subject to an acceleration, constant or not, whose direction is continually perpendicular to the direction of motion, the velocity will not be alteredand the only effect of the acceleration will be to make the point move in a curve whose curvature is proportional to the acceleration at each instant, and inversely as the square of the velocity.

89. In other words, if a point move in a curve, whether with a uniform or a varying velocity, its change of direction is to be re garded as constituting an acceleration towards the centre of curva→→ ture, equal in amount to the square of the velocity divided by the radius of curvature. The whole acceleration will, in every case, be the resultant of the acceleration thus measuring change of direction, and the acceleration of actual velocity along the curve.

40. If for any case of motion of a point we have given the whole velocity and its direction, or simply the components of the velocity in three rectangular directions, at any time, or, as is most commonly 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.

(b) 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 (S$ 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 Ꮴ has the constant value. See § 36.

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 =o the velocity was V, then at time f it is

v=V+at.

Also the space (x) described is equal to the product of the time by the average velocity. But we have just shown that the average velocity is = } (V+V + at) = V +}at,

and therefore

Hence, by algebra,

or

xVt+ fat.

V3 + 2ax =V® + 2 Vat+ d°fa = (V+at}' = v3,
{ v1- {V!=ax.

If there be no initial velocity our equations become

v=at, x=}ata®; }va=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 hat is described parallel to the direction of acceleration. Hence, if AP be the direction of motion at A, AB the direction of acceleration, and Q the position of the point at time, t; draw QP parallel to BA, meeting AP in P: then

C

P

and is therefore known. Also QA 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, whatever (and however varying) be its magnitude, 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.

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; 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 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 parallelogram; and therefore the remainder, OAD, is equal to the sum of the two half-moments. But OAD is half the moment of the resultant velocity round the point O. 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 O is within the angle CAB.

B

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,