to the axis of x, the angle X,PX2=X,PF. It follows that all rays

of light or heat, or of sound, parallel to the axis of the parabola, will be collected in F.

parabola.

Hence F is denominated the Focus of the

Cor. 1. The points where the normal intersects the para- (238) bola and its axis, are equally distant from the focus, and the normal is consequently equally inclined to the axis and to the radius vector, terminating in the same point of the curve; ...

Cor. 2. The tangent makes equal angles with the axis (239) and the line joining the point of tangency and the focus. In equation (232) the abscissa of the focus is p;

Scholium I. The method here employed for drawing a tangent to the parabola is obviously applicable to all curves; and it is recommended to the student to make himself familiar with it by drawing tangents to the circle, ellipse and hyperbola, and to verify his results by the properties already demonstrated in regard to these

curves.

Scholium II. We must not pass unnoticed the remarkable symbol, by which we have readily arrived at important relations. Since in we have reduced both h and k to zero, it is natural to regard 0 this expression as equivalent to and this in itself, abstractly considered, has no meaning at all, for to it we cannot attach any idea

SCHOLIUM.

independent of its origin. But to regard the symbol []

h

125

as des

titute of signification, or not indeed as possessing an important one, would be to attribute to it an altogether erroneous interpretation. In truth, it not only indicates a quantity, but that quantity as evolved, by a peculiar operation, from specific conditions.

The symbol [] signifies,

1o. There are two quantities which are regarded as variable, y and x.

2o. y is regarded as depending upon x.

3o. Increments [increases], k and h, are given to these variables, y, x.

4°. The ratio,

k

h of the increment of y to that of x is found.

obtained by diminishing h, and consequently k, to zero.

It is also to be observed that the ratio,

k

[]

will generally it

" h

self be a variable quantity. Indeed, in this particular case of a

If a curve be such that the distance of any point of (241) it to a point fixed in space shall bear a constant ratio, e, to the distance of the same point of the curve to a given line or Directrix, then will the curve be either the Ellipse, Hyperbola or Parabola, according as the ratio, e, is less than, greater than, or equal to unity, [e ≥ 1].

Let the fixed line or directrix be the axis of Y y, and the perpendicular to it drawn through the fixed point F, the axis of x, and let the distance of F from the origin be denoted by d; there results,

y2+(d-x)2 = 22 = e2x2, since

2

--

Ꮖ

=e, by hypothesis ;

.._y2+(1 − e2)x2 — 2dx + d2 = 0, which becomes at once

the equation of the parabola, when e = 1, or (1 — e2) = 0. In order to make the term 2dx, affected by the first power of

x, disappear, we will transport the origin to the right a distance =m, so that we shall have x + m

instead of x,

or

••. y2 + (1 − e2) (x + m)2 — 2d(x + m) + d2 = 0,

y2 + (1 − e2)x2 + [(1 − e2) • 2m — 2d]x+(1

— e2)m2 — 2dm + d2 = 0,

m

Fig. 492.

from which, attributing such a value to m as to make the term affected by the first power of r disappear, we have

and

y2+(1 − e2) x2 + (1 − e2) m2 - 2dm+d2=0;

according as 1- e2 is +, or —, that is, e < 1, or e> 1. Q. E. D.

Scholium. It is to be observed that the Ellipse, Circle, Hyperbola, and Parabola may be represented by the same general equation, and are therefore to be regarded as nothing more than species of the same curve.

10. From the top of a tower 48 feet high, a cannon ball is fired in a horizontal direction with a velocity of 1000 feet per second. Required the distance from the foot of the tower where the ball will strike the horizontal plane on which it stands; no allowance being made for atmospheric resistance, and the vertical descent being in the times 1, 2, 3, &c., seconds, 12. (16), 22. (16), 32. (16), &c., feet.

2o. To transport the origin to a point of the Parabola, the new axes being parallel to the old.

Let b, a, be the coördinates of the new origin referred to the old axes; we have

3o. To ascertain whether a board cut from a log next the roots without having been squared, may be regarded as an inverted parabola.

Let the middle line of the board be taken for the Y axis of x and the broader end for the axis of y; the preceding problem gives us

(cy)2+2b(cy) = 2px,

Fig. 51.

where there are three constants, b, c, p, to be determined, one of which, c = half the width of the broader end, may be supposed known. We must therefore determine the values of b and p from values of x and y taken in two different places, and then see if b and p remain the same, or nearly the same, for measures taken throughout the length of the board.

4°. The length of a board, of the form given in 3°, is 8 feet, the ends are 4 and 2'4 ft. broad, and the breadth of the middle is 3 ft. Required the equation of its edge, the axes of coördinates being as in the last. Ans. (2―y)2+ ‘7(2− y) = '15x, or y247y='15x-5'4.

5°. It is required to form a gauge by which to turn a parabolic mirror 18 inches in diameter, and having a focal distance of 10 inches, measuring from the diameter. Required the depth of the mirror and its equation.

Ans. Depth 1'7268. Equation, y2 = 46'9072. x.

6°. Fold the lower corner of the left hand page, so as to make the area of the folded part constantly equal to a given square (a2) the side of which is a; and find the locus of the corner, or the curve in which it is constantly situated.

If we take the position of the corner before folding for the origin of coördinates and the two edges of the leaf for the axes, we shall find

(y2 + x2)2 = 2 (2a)2yx,

for the equation sought; from which it appears that the curve is symmetrical in reference to the axes or edges of the leaf, and that it begins and ends at the origin, since x = 0 gives y = 0.