When a cone is cut by a plane that makes with the axis an angle that is less than a right angle, but not so small an angle as the angle which the side of the cone makes with it, such section is an ellipse.

A section of a cone making an angle with the axis equal to that which the side makes is a parabola.

A section of a cone which makes, with the axis, an angle that is less than that which the side makes is an hyperbola.

A section of a cone with which the axis coincides is an isosceles triangle.

432. Cut from an apple or a turnip as accurate a cone as you can, and give a specimen of each of the five conic sections.

433. Give a sketch of a builder's trammel, and make an ellipse with a trammel; and show that you can, on the principle on which it acts, make an ellipse without one.

The long diameter of an ellipse is called the axis major, and the short one the axis minor, and the distance of either of the foci of the

ellipse from its centre is called the eccentricity

of the ellipse.

434. Give a figure to show what is meant.

Two lines drawn from the foci of an ellipse to a point in the circumference make equal angles with a tangent to the ellipse at that point.

435. Can you at a particular point in the circumference of an ellipse draw a tangent to that circumference ?

With a pair of compasses, and using differ ent-sized circles, how nearly can you imitate an ellipse? In other words

436. How would you make an oval?

437. Can you, out of a circular piece of ma hogany, and without any loss, make the tops of two oval stools, with an opening to lift it by, in the middle of each?

The solid formed by revolving on its minor axis a semi-ellipse is called an oblate spheroid.

438. Show how an oblate spheroid is formed, and say what the oblate spheroid reminds you of

The solid formed by revolving on its axis major one-half an ellipse is called a prolate spheroid.

439. Show how a prolate spheroid is formed. and say what it reminds you of.

440. Supposing a room to be built in the form of a prolate spheroid, and a person to speak from one focus, show where his voice would be reflected.

441. Would there be the same effect produced in a room built in the form of an oblate spheroid.

Provided no notice is taken of the resistance of the air, a stone thrown horizontally from the top of a tower, at a velocity of 48 ft. in a second, and subject to the incessant action of the earth, which from nothing induces it to fall by a uniformly-increasing velocity through about 16 ft. in the first second, 48 ft. in the second second, 80 ft. in the third second, 112 ft. in the fourth second, and so on, makes in its progress a kind of curve. Now, the terins of the series 16, 48, 8υ, 112, 144, etc., increase in

a certain ratio; and, if 16 be called 1, 48 will be 3, 80 will be 5, 112 will be 7, and 144 will be 9, etc. These distances may then be expressed as falling distances, thus, 1, 3, 5, 7, 9, etc. And, keeping in mind that the horizontal velocity remains uniform, that is 48 ft., i. e., 3 × 16 ft. in a second, we have two kinds of dimensions at right angles to each other, from which to make the curve. This curve is called a parabola.

442. Can you construct a parabola?

When these distances, instead of being writ ten down as the separate result of each second's action, are successively added to show the combined results, we have for

From this it will be seen that the distance

fallen is as the square of the time, i. e., in

seconds the distance fallen will be 6a× 16 ft. = 36 × 16 ft. = 576 ft.

443. Required the distance a stone falls in half a second.

444. Required the distance a stone falls in 21 seconds.

445. Can you show that there are two kinds of quadrilaterals in which the diagonals must be equal, two kinds where they may be equal, and two kinds where they cannot be equal?

446. Can you make in card a tetrahedron whose four surfaces shall be unlike in form ꞌ

THE END.