CENTRES OF CURVES.

44. To find the centre of any curve of the second order.

Definition. The centre of a curve is a point such that every secant passing through it meets the curve in two points which are equidistant from the point in question.

It follows immediately, from this definition, that if the origin of coordinates be placed at the centre of a curve of the second order, the equation of the curve will be such as to give equal values of x when y = 0, or equal values of y when x = 0; that is, the equation will not contain the first powers of x and y. This property furnishes a simple method of finding the centres of such curves from their equations. Thus taking the equation (1) of the ellipse (Art. 40), viz. :—

remove the origin to a point (a, B), by writing + a for x and y +ẞ for y; then the equation becomes

a2 (y+B)2 = b2 (2 ax+2aa-x2 - 2 α x-a3), or a2 y2 + b2x2 + 2 a2 By+2b2 (a− a) x = b2 (2a a a2) - a2 ß2. Now in order that (a, B) may be the centre, the coefficients of x and y in this equation must vanish;

a2 y2 + b2x2 a2 b2,

=

which is the same as the equation (2) of the ellipse (Art. 40) when the centre is the origin. And similarly for the hyperbola.

Next in the equation of the parabola (Art. 38), write x+a for x and y+B for y;

then (y+B) = 4m (x + α), or y2 + 2 By + B2 = 4mx + 4 ma. Now the coefficient of x cannot vanish, for then m = AS = 0, and consequently, the parabola has not a centre.

DIAMETERS OF CURVES.

45. To find the equation of a diameter to any curve of the second order.

Definition. The diameter of a curve is the locus of the middle points of a series of parallel chords.

(1),

be the equation of a parabola, the vertex being the origin (Art. 38), and

y = px + q

. (2),

the equation of a given straight line; then (Art. 12) the equation

y = px + q' . . . . (3),

represents any straight line parallel to (2), q' admitting of all possible values.

Eliminating y between (1) and (3), there results for the points of intersection of the parabola (1), and the line (3), the equation

q'2
= 0

p2

(4).

Hence if x, x be the roots of this equation, and x', y' the coordinates of the middle point of the chord which joins the points of intersection of (1) and (3), we have, by Art. 23, and Art. 125, Algebra,

Eliminating q' which varies for different positions of the parallel chords, between (5) and (6), there results for the locus of the middle points of the parallel chords, the equation

y': , or (suppressing the dash) y =

=

2m Р

2m

(7),

which (Art. 12) is a straight line parallel to the axis of x. Hence all diameters of the parabola are parallel.

Similarly, the equation of a diameter to the ellipse

is the analogous equation for the hyperbola.

Hence all the diameters of the ellipse and hyperbola pass through the

centre.

Cor. 1. Let a diameter of the parabola (1) pass through the point (h, k) in the curve. Then because it is parallel to the axis of x its

equation (Art. 12) is

be one of the chords which this diameter bisects, its equation by (7) is also

Comparing this with the equation of the tangent at the point (h, k), Art. 43, it is easily seen that the tangent and the chord (13) are parallel; and similarly for the ellipse and hyperbola. Hence the chords bisected by any diameter are parallel to the tangent at the extremity of that diameter.

Definition. Two diameters of the ellipse and hyperbola so related that each bisects the chords parallel to the other, are called conjugate diameters.

Cor. 2. Since the angular ccefficients of the equations of tangents to

the ellipse and hyperbola at the point (h, k), Art. 43, remain the same when hand-k are written for h and k, it follows that the tangents at the extremities of any diameter are parallel.

MISCELLANEOUS EXERCISES ON THE CONIC SECTIONS AND OTHER

CURVES.

1. Prove that if a circle be described on the radius vector of a parabola, the tangent at the vertex is a tangent to the circle.

2. Find the condition that the line y = ax + 8 may be a tangent to the parabola y2 = 4mx. Ans. aß m.

=

3. Show how you can find all the geometric elements of a conic section (focus, vertex, directrix, etc.) from a given portion of its arc.

4. Find the locus of the centre of a circle which constantly touches a fixed straight line, and passes through a given point not in the line. - Ans. A parabola.

5. Prove that the locus of the centre of a circle which touches a given straight line and a given circle is a parabola.

6. Prove that if P be any point in an ellipse whose foci are S and H,

7. The length of the perpendicular upon the tangent from the centre of an ellipse is equal to a (1 − e2 cos2 )*, in which is the inclination of the tangent to the major axis.

8. Prove that the locus of the foci of all the parabolas having the same directrix, and a point common to all, is a circle.

9. If in an ellipse there be taken three abscissas in arithmetical progression, the radii vectores drawn from the focus to the extremities of the ordinates at those points will also be in arithmetical progression.

10. What is the locus of the vertices of parabolas having the same tangent and the same directrix? Ans. A straight line.

11. Given the major axis of an ellipse, and also a fixed point on the minor axis, from which a normal is drawn to the curve, what is the locus of the points of intersection of the normal and ellipse?

Ans. The locus is a circle.

12. Prove that two tangents to a parabola drawn from the same point in the directrix are at right angles to each other.

13. Prove that the tangent at any point of a parabola meets the directrix and latus rectum produced in two points equally distant from the focus.

14. Draw in a parabola a chord of given length, and find the locus of its middle point. Ans. A curve of the fourth order. 15. At what point in the ellipse a2 y2 + b2x2 = a2 b2 does the tangent make an angle of 45° with the radius vector drawn to the same point?

16. Find the condition that two tangents drawn from the same point

(a, b) to the parabola y = 4m x, may be equal.

Ans. B = 0.

17. Let A CB be a semicircle of which A B (2 r) is the diameter, BD an indefinite straight line perpendicular to AB, and ACD a straight line meeting the semicircle in C and BD in D; then if P be a point in AC such that AP = CD, the locus of P is the Cissoid of Diocles. Find its equation, and thence show that BD is an asymptote to the curve. Ans. The equation is y' (2rx) — x3 = 0. 18. The Conchoid of Nicomedes is thus generated: A B is an indefinite straight line, and C a given point without it; from C a perpendicular CDP is drawn to AB meeting it in D, and straight lines C D' P', C D" P", etc., are drawn in such manner that DP, D'P', D" P", etc., are all equal, D', D', etc., being in A B, then the locus of the points P, P', etc., is the Conchoid. Find its equation, when CD = a, and DP = b, and thence trace the curve. Ans. x2 y2 = (b2—y2) (a+y)3. 19. If MQ be an ordinate to the semicircle A QB (radius = r), and it be produced to P so that MP: MQ::AB: AM; then the locus of P is a curve called the Witch of Agnesi. Find its equation, and thence deduce some of its geometrical properties.

Ans. Its equation is y2 x = 42 (2r-x).

ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS.

I. THE DIFFERENTIAL CALCULUS.

ELEMENTARY PRINCIPLES AND ILLUSTRATIONS.

1. THE algebraic processes which have been already investigated enable us to solve a large number of the questions which arise in practice, but for the solution of others they are inconvenient, and often inadequate. In the annexed figure, let A B represent a curved line, supposed to be described in accordance with some given law, by which its distance from the straight line C D is constantly varied. It may be desirable to determine the point in CD at which the distance, as E F, is the greatest, or as G H,

A

E

с

H

D

the least, and the measure of these distances, and also to determine the area of E FG H, and consequently the side of a rectangle which would give the same area, or the measure of the average distance between the two lines; but to effect these, in every case, exceeds the power of algebra, and we must have recourse to a new calculus adapted to inquiries of this nature.

The difference between the case which is here assumed and the cases to which the usual processes of algebra are applicable, may be easily traced. By these processes we are enabled to ascertain the values of particular quantities which are dependent on each other, when as many distinct conditions with reference to some known quantity or quantities are given, from which as many equations can be obtained as there are unknown quantities, while the cases for which a new calculus is required appear to be those in which variable quantities, as the ordinate F E, and the area E F G H, dependent on the value of another variable quantity, as C F, have to be dealt with. The general principles on which a calculus adapted to such investigations has been formed will be explained presently.

2. The study of algebra leads to that of variable quantities, in which a letter is considered as assuming all possible values between given limits, and the results of that supposition are developed in the differential and integral calculus in which all quantities are considered either constant or variable.

A constant quantity is one whose value or magnitude remains unchanged throughout the operation or investigation in which it is employed.

A variable quantity is one which admits of an unlimited number of