CALCULATION OF AN AREA.

71

this supposition makes the velocity too great, excepting at the instants a, b, c...; therefore the actual distance traversed is less than the area of the outer staircase.

It appears therefore that the distance traversed in the actual motion is represented by an area which lies between the area of the outer and the area of the inner staircase. But the area of the curve of velocities lies between these two. Therefore the difference between the area of the curve of velocities and the area which represents the distance traversed is less than the difference between the areas of the outer and inner staircases. Now this last difference is less than a rectangle, whose height is rv and whose breadth is the greatest of the lengths ma, ab, ...; for it is made up of all the small rectangles on the curve like ufAg. But we may divide the interval mn into as many pieces as we like, and consequently we may make the largest of them as small as we like.

It follows that there is no difference between the area of the curve of velocities and that which represents the distance traversed. For if there is any, let it be called 8. Divide mn into so many parts that a rectangle of the height nv, standing on any of them, shall be less than this area S. Then we know that the difference in question is less than a rectangle of height rv standing on the greater of these parts, that is, less than 8; which is contrary to the supposition.

This demonstration indicates a method of finding the area of a curve, and, at the same time, of finding the distance traversed by a point moving with given velocity. The method is the same in the two problems (which, as we have just seen, are really the same) but has to be described in somewhat different language. For the area of the curve umnv, the rule is: Divide mn

into a certain number of parts, and on each of these erect a rectangle whose height is the height of the curve at some point vertically over that part; then the sum of the areas of these rectangles will differ

from the area of the curve by a

Y

X

quantity which can be made as small as we like by increasing the number of parts and diminishing the largest of them. For the distance traversed during a certain interval, the rule is: Divide the interval into a certain number of parts, and suppose a body to move uniformly during each of those parts with a velocity which the actual body has at some instant during that part of the interval; then the distance traversed by the supposed body will differ from that traversed by the actual body by a quantity which can be made as small as we like by increasing the number of parts and diminishing the largest of them.

For example (Wallis), suppose the velocity at time t to be th, and that we have to find the space described in the interval from t=a to t=b. Let this interval be divided into n parts in geometric progression, as follows. Let o"=b: a, so that boa. Then the parts shall be the intervals between the instants a, oa, o a,...o"1a, b. The velocities of the moving body at the beginnings of these intervals are ak, okak, o2kak,...o(n-1)kak, bk... Hence if a body move uniformly through each interval with the velocity which the actual body has at the beginning of that interval, it will describe the space

ak (oa − a) +okak (o3a − σạ) + ... + o(n−1)% ak (ona — on-1a)

= ak+1(σ − 1) (1 + σk+1 +σ2(k+1) + ... +σ(n−1)(k+1))

=ak+1(σ-1)

n(k+1)_1
+1-1

=

bk+1 - ak+1 1+o+o+...+ok.

2

Now the larger n is taken, the more nearly σ approaches to unity, and consequently, the more nearly the denominator of this fraction approaches to the value k+1. Thus the space described from t=a to t=b is (b+ - ak+1): k+1. By making a=0 and b-t in this formula, we find that the space traversed between 0 and t is t+1 k+1. This agrees with our previous investigation; for if (k+1) s=tk+1, we know that stk. As in the converse investigation, p. 55, it is easy to extend the method to the case in which k is a commensurable fraction; for the quotient σk+1—1: σ −1 approaches also in that case the value k+1 when σ approaches unity.

DEFINITION OF INTEGRAL.

73

As an example, we may find the area of a parabola. Here pn2 varies as an, or pn = μ. an3. Thus we must put k=. Then area abc=μ. ab, but μ. ab

. ab3 = bc,

ab. bc = two

thirds of the circumscribing rectangle abcd.

A small interval of time being denoted by St, the approximate value of s is the sum of a series of terms like sot, which we may write Σsst. The value to which this sum approaches when the number of intervals St is increased and their size diminished, is written fsdt. Thus the equation s= fsdt is shorthand for this statement: s is the value to which the sum of the terms sdt approaches as near as we like when the number of the St is increased and their size diminished sufficiently. When the whole interval considered lies between ta and tb, we indicate these limits of the interval thus: sf sdt. This expression is called the integral of s between the limits a and b, or from a to b. Observe that although the sign f takes the place of Σ, and sdt of sot, yet f does not mean sum, nor sdt a small rectangle of breadth dt and height s. The whole expression must be taken as one symbol for a certain quantity, which indicates in a convenient way how that quantity may be calculated. fsdt is the value to which the sum Σsst approaches; it is not itself a sum, but an integral, that is to say, a quantity which may be approximately calculated as the sum of a number of small parts.

The result obtained on the previous page may now be written thus:

f tkdt=tk+1: k +1.

CURVATURE.

A plane curve may be described by a point and a straight line which move together so that the point always moves along the line and the line always turns round the point. (Plücker.) Let s be the arc of the curve,

measured from a fixed point a up to the moving point p, and let o be the angle which the moving line (the tangent)

makes with a fixed line. Then the linear velocity of the point along the line is s, and the angular velocity of the line round the point is . The ratio : is called the curvature of the curve at the point p. This ratio is the

s-flux of ; for we know that, since & is a function of s which is a function of t, s. p, see p. 66. Thus we s., may define the curvature as the rate of turning round per unit of length of the curve.

We may also define it independently of the idea of velocity, thus. The angle between the direction of the tangents at a and b is called the total curvature of the arc ab; the total curvature divided by the length of the arc is called the mean curvature of the arc; and the curvature at any point is the value to which the mean curvature approaches as nearly as we like when the two ends of the arc are made to approach sufficiently near to that point.

In a circle of radius a, the arc sap; consequently s=ap, and ☀ : s=1 : a, or the curvature is the reciprocal of the radius. (Observe that curvature is a quantity of the dimensions [L].) It is in fact obvious that the arc of a small circle is more curved than that of a large

one.

When the point stops and reverses its motion, while the line goes on, we have a cusp in the curve; at such a point 80, while & is finite, and the curvature is infinite. When the line stops and reverses its motion, while the point goes on, we have a point of inflexion; at such a

=

CIRCLE OF CURVATURE.

75

point 0 while s is finite, and the curvature is zero. When both motions are reversed, we have a rhamphoid

기

1

cusp or node-cusp; the curvature is in general finite and the same on both branches.

A circle touching a curve and having the same curvature on the same side at the point of contact is called the circle of curvature at that point. Its radius is called the radius of curvature. Its centre is called the centre of curvature. In general the curvature is greater than that of the circle

P

on one side of the point, and less on the other; so that the curve crosses the circle, passing outside where its curvature is decreasing and inside where it is increasing. If then we draw a circle to touch a curve at a point p and cut it at a point q, and then alter the radius of the circle, by moving the centre o along the normal at p, until q moves up to p, we shall obtain the circle of curvature. Hence also this circle may be described as one which has three points of intersection combined into one point; for the contact at p already combined two points.

At a point of maximum or minimum curvature (like the ends of the axes of an ellipse) the curve lies wholly inside or wholly outside the circle, as in a case of ordinary contact; in such a case four points of intersection are combined into one.

We know that (p being the position-vector op) p is the velocity, and is therefore parallel to the tangent at p; and