2. Add 4 times the square root of the product of the transverse and abscissa, to the product last found, and divide the sum by 75.

3. Divide 4 times the product of the conjugate and abscissa by the transverse, and this last quotient multiplied by the former will give the area required nearly.

EXAMPLES.

In the hyperbola GAF, the transverse axis is 30, the conjugate 18, and the abscissa or height AH is 10; what is the area?

=c=conjugate axis, by the nature of the

And this thrown into a series will very nearly agree with the former; which shows the rule to be an approximation. Q. E. I.

Rule 2. If 2x, 2y=bases, v, and v their distances from the centre, and the other letters as before, then will vy—

Rule 3. If t be put = transverse axis, c = conjugate, and x = abscissa, the area of a segment of an hyperbola, 4√tx+4x2+√ tx

ut off by a double ordinate will be =

15

=

F

Here 21 √(30 × 10+ × 102) — 21 √300 + 500 ÷ 7 = 21 √300 + 71.42857-21 √371.42857 =21 × 19.272= 404.712.

And (4/30x 10+404.712)÷75=(4√300+404.712) ÷75=(4× 17.3205+404.712)÷75=(69.282+404.712)

÷75-473.994÷75-6.3199.

× 4× 6.3199=24× 6.3199=151.6776=area required. 2. The transverse diameter is 100, the conjugate 60, and the less abscissa 50; what is the area of the hyperbola? Ans. 3220.363472.

3. Required the area of the hyperbola to the abscissa 25, the two axes being 50 and 30.

Ans. 805.0909.

OF THE

MENSURATION OF SOLIDS.

DEFINITIONS.

1. The measure of any solid body, is the whole capacity or content of that body, when considered under the triple dimensions of length, breadth, and thickness.

2. A cube whose side is one inch, one foot, or one yard, &c. is called the measuring unit; and the content or solidity of any figure is computed by the number of those cubes contained in that figure.

3. A cube is a solid contained by six equal square sides.

4. A parallelopipedon is a solid contained by six quadrilateral planes, every opposite two of which are equal and parallel.

5. A prism is a solid whose ends are two equal, parallel, and similar plane figures, and whose sides are parallelograms.

Note.-When the ends are triangles it is called a triangular prism; when they are squares, a square prism; when they are pentagons, a pentagonal prism, &c.

6. A cylinder is a solid described by the revolution of a right angled parallelogram about one of its sides, which remains fixed.

7. A *pyramid is a solid whose sides are all triangles meeting in a point at the vertex, and the base any plane figure whatever.

Note.-When the base is a triangle, it is called a triangular pyramid; when a square, it is called a square or quadrangular pyramid; when a pentagon, it is called a pentagonal pyramid, &c.

8. A sphere is a solid described by the revolution of a semicircle about its diameter, which remains fixed.

9. The centre of a sphere is a point within the figure, everywhere equally distant from the convex surface of it. 10. The diameter of the sphere is a straight line passing

* The definition of a cone has been given already.

through the centre, and terminated both ways by the convex superficies.

11. A circular spindle is a solid generated by the revolution of a segment of a circle about its chord, which remains fixed.

12. A spheroid is a solid generated by the revolution of a semi-ellipsis about one of its diameters, which is considered as quiescent.

The spheroid is called prolate, when the revolution is made about the transverse diameter, and oblate when it is made about the conjugate diameter.

13. Elliptic, parabolic, and hyperbolic spindles, are generated in the same manner as the circular spindle, the double ordinate of the section being always fixed or quies

cent.

14. Parabolic and hyperbolic conoids, are solids formed by the revolution of a semi-parabola or semi-hyperbola about its transverse axis, which is considered as quiescent.

15. The segment of a pyramid, sphere, or of any other solid, is a part cut off from the top by a plane parallel to the base of that solid.

16. A frustrum or trunk, is the part that remains at the bottom, after the segment is cut off.

17. The zone of a sphere, is that part which is inter