scalar magnitude r and 'respectively, there will be two points corresponding to any two given values of r and r'; namely P and P' the intersections of the two circles with centres at A and B and radii equal to r and rrespectively. We may distinguish these points as being one above, and the other below A B. Only in the case of the circles touching will the two points coincide; if the circles do not meet, there will be no point.

If P moves so that for each of its positions with regard to A and B the quantities r and r' satisfy some definite relation, we shall obtain a continuous set of points in the plane or a curved line of some sort. For example, if we fasten the ends of a bit of string of length l to

pins stuck into the plane of the paper at A and B, and then move a pencil about so that its point P always remains on the paper, and at the same time always keeps the string APB taut round its point, the pencil will trace out that shadow of the circle which we have called an ellipse.

In this case r+r=AP+PB = 1, the constant length of the string. This relation r+rl is an equation between the scalar quantities r, r' and l, which holds for every point on the ellipse, and expresses a metric property of the curve with regard to the points A and B.

А

If on the other hand we cause P to move so that the difference of AP and BP is a constant length (r-r=), then P will trace out the curve we have termed the

hyperbola. We can cause P to move in this fashion by means of a very simple bit of mechanism. Suppose a rod B L capable of revolving about one of its ends в let a string of given length be fastened to the other end L and to the fixed point A. Then if, as the rod is moved round B, the string be held taut to the rod by a

P

B

FIG. 61.

pencil point P, the pencil will trace out the hyperbola. For since LP+PA equals a constant length, namely that of the string, and LP+PB equals a constant length, namely that of the rod, their difference or PA-PB is equal to the constant length which is the difference of the string and the rod.

Α

E

FIG. 62.

The points A and B are termed in the cases of both ellipse and hyperbola the foci. The name arises from the following interesting property. Suppose a bit of polished watch spring were bent into the form of an ellipse so that its flat side was turned towards the foci of the ellipse; then if a hot body were placed at one focus B, all the rays of heat or light radiated from B

which fell upon the spring would be collected, or, as it is termed, focussed' at A; hence A would be a much brighter and hotter point than any other within the ellipse (B of course excepted). The name focus is from the Latin, and means a fireplace or hearth. This property of the arc of an ellipse or hyperbola, that it collects rays radiating from one focus in the other, depends upon the fact that AP and PB make equal angles with the curve at P. This geometrical relation corresponds to a physical property of rays of heat and light; namely, that they make the same angle with a reflecting surface when they reach it and when they leave it.

A third remarkable curve, which is easily obtained from this our first method of considering position, is the lemniscate of James Bernoulli (from the Latin lemniscus, a ribbon). It is traced out by a point p which moves so that the rectangle under its distances from a and B is always equal to the area of a given square

3

FIG. 63

(r.c2). If the given square is greater than the square on half A B, it is obvious that P can never cross between A and B; if it is equal to the square on half A B, the lemniscate becomes a figure of eight; while if it is less, the curve breaks up into two loops. In our figure a series of lemniscates are represented. A set of curves obtained by varying a constant, like the

given square in the case of the lemniscate, is termed a family of curves. Such families of curves constantly

occur in the consideration of physical problems.

direction is A B. angle BAP, we position of P. AP and

§ 6. Polar Co-ordinates.

(B) The points A and B determine a line whose If we know the length A P and the shall have a means of finding the Let r be the number of linear units in the number of angular units in BA P, where may of course be fractions. In measuring the angle we shall adopt the same convention as we have employed in discussing areas (see p. 134); namely, if a line at first coincident with AB were to start from

r and

that position, and supposed pivoted at A to rotate counter-clockwise till it coincided with AP, it would trace out the angle 0. Angles traced out clockwise will like areas be considered negative. Thus the angle BAP' below AB would be obtained by a rotation clockwise from AB to AP', and must therefore be treated as negative. On the other hand, we might have caused a line rotating about A to take up the position A P' by rotating it counter-clockwise through an angle marked in our figure by the dotted arc of a circle. Further we

might obviously have reached AP by a line rotating about a clockwise, and might thus represent the position of p by a negative angle. But even after we had got to P we might cause our line to rotate about a a complete number of times either clockwise or counter-clockwise, and we should still be at the end of any such number of complete revolutions in the same position A p.

We have then the following four methods of rotating a line about A from coincidence with AB to coincidence with A P:

(i) Counter-clockwise from a B to A P.

(ii) Clockwise from A B to A P.

(iii) The first of these combined with any number of complete revolutions clockwise or counter

clockwise.

(iv) The second of these combined with any number of complete revolutions clockwise or counter

clockwise.

The following terms have been adopted for this method of determining position in space:

The line A B from which we begin to rotate our line is termed the initial (beginning') line; the length AP is termed the radius vector (from two Latin words signifying the carrying rod or spoke, because it carries the point p to the required position); the angle BAP is termed the vectorial angle, because it is traced out by the radius vector in moving from A B to the required position A P; A is termed the pole, because it is the end of the axis about which we may suppose the spoke to turn. Finally AP (= r) and the angle BAP (= 0) are termed the polar co-ordinates of the point P, because they regulate the position of P relative to the pole a and the initial line A B.