Whence 675+66.6536 ÷ 675+45.0216 = 741.6536÷ 720,0216=1.03004; and 1.03004 × 10=10.3004=length of the arc required.

2. The transverse diameter of an hyperbola is 120, the conjugate 72, the ordinate 48, and the abscissa 40: required the length of the curve. Ans. 62.6496. 3. Required the whole length of the curve of an hyperbola, to the ordinate 10; the transverse and conjugate axes being 80 and 60. Ans. 20.6006.

PROBLEM XIV.

To find the area of an hyperbola, the transverse, conjugate, and abscissa being given.

RULE.*

1. To the product of the transverse and abscissa, add 4 of the square of the abscissa, and multiply the square root of the sum by 21

* Demon. Let t=transverse diameter, c=conjugate, x=

bscissa, y=ordinate, and = Then it is well known

z=;

that 4xy × (-1.3.5. —

1

3.5.7.

-z3, &c.) =

5.7.9.

area of the hyperbola.

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 2v, 2y=bases, v, and v their distances from the centre, and the other letters as before, then will vy

and abscissa, the area of a segment of an hyperbola

ut off by a double ordinate will be

15

=

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

And (430x 10+404.712)÷75 (4√300+404.712) ÷75 (4× 17.3205+404.712)÷75=(69.282+404.712)

× 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 23, the two axes being 50 and 30. Ans. 805.0909.

6*

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.