sector of these solids are as the ratio of rr', as has been shown with regard to the convex surface. The cone whose base is the base of the segment, and whose vertice is the centre of the polyedroid or sphere,formula (6, Art. 9,) may be expressed 2arah 3πrh2+h3

-

3

hence the segment will be

r22h — r2h+«rh2 —«h3 =«h({r'+{r*+rh — h3)

(6.)

If the polyedroidal segments consists only of a vertical cone, its solidity will be 2rh-h')}h=«rh2 — }«h', subtract this from the spherical segment on the same base, formula (7, Art. 9,) xrh'h', and we have rh3. (7.)

which is the value of that portion of the segment of the sphere not included in that of the inscribed polyedroid, which is such a portion of the sphere as would be generated by the revolutions of a circular segment as BD, about the axis DC, passing through the centre of curvature, and perpendicular to the arc of the segment at the point of contact D.

A

PROBLEM.

It is required to find when the spherical segment and the cone composing a spherical sector are equal to each other.

Let ABED represent a sphere generated by the revolution of a semicircle ABE about its diameter AE. The sector ABC, by this revolution, generates a spherical sector, which is composed A of a spherical segment generated by the revolution of a semisegment ABP, and of a cone generated by the revolution of the right-angled triangle BPC.

E

The solidity of the sector, formula (3, Art. 9.) will be «r2h.

Th
3

The solidity of the cone, rh«rh2+, formula (6.)

Now, in order that the cone may be equal to the segment, the sector, which is the sum of both, must be double the cone: hence, rh=f«r2h=2«rh2+3«h', dividing by 2, transposing, &c.,

Of these two solutions it is evident that only h=r-V can satisfy the conditions of the question, since r+V123 is more than 2r, or more than the diameter of the sphere.

EXAMPLES FOR EXERCISE.

1. What is the solidity of the spherical segments of which the frigid zones are the convex surfaces, the altitude of each segment being 327 miles, and the radius of the base 1575,28 miles? Ans. 1282921583 solid miles nearly.

2. What is the solidity of the spherical segments of which the temperate zones are the convex surface, the radius of the superior base being 1575,28 miles, that of the inferior 3628,86 miles, and the altitude 2053,7 miles?

Ans. 55021192817 solid miles nearly. 3. What is the solidity of the spherical segment of which the torrid zone is the convex surface, the radii of the bases being 3628,86 miles, and the altitude 3150,6 ?

Ans. 146715018499 solid miles nearly.

4. Having two vats or two tubs in the form of conical frusta, whose dimensions are as follows, viz. the first has a base whose diameter is 3 feet, its altitude is 31 feet, and the slant height of its side is 4 feet; the diameter of the base of the second is 3 feet, its altitude is 5 feet, and the curve surface is 60 square feet, what must be the dimensions of one capable of containing as much as the other two, if the diameter of the bottom and top, and the altitude are in the proportion of 2 21 and 3.

5. What is the difference in surface of a vertical hexedroid circumscribing a sphere whose diameter is 10, and the whole surface of a conesected frustum of a cone inscribed in the same sphere, and whose wanting base is 6, and perfect base 4?

CONIC SECTIONS.

There are three curves, whose properties are extensively applied in mathematical investigations, which, being the sections of a cone made by a plane in different positions, as will be shown in another place, are called the Conic Sections. These are,

1. The Parabola. 2. The Ellipse. 3. The Hyperbola.

PARABOLA.

DEFINITIONS.

1. A Parabola is a plane curve, such, that if from any point in the curve two straight lines be drawn; one to a given fixed point, the other perpendicular to a straight line given in position: these two straight lines will always be equal to one another.

2. The given fixed point is called the focus of the parabola. 3. The straight line given in position, is called the directrix of the parabola.

Thus, let QAq be a parabola, S the focus, Nn the directrix;

Take any number of points, P,, P2, P3,

in the curve;

39

Join S, P, ; S, P, ; S, P ̧'; ... and draw P, N,, P, N,, P, N.

2

to the directrix; then

39....

N

NI

N2

perpendicular

SP, P, N,, SP,=P,N,, SP,= P,N,,...

P

4. A straight line drawn perpendicular to the directrix, and cutting the curve, is called a diameter; and the point in which it cuts the curve is called the vertex of the diameter.

5. The diameter which passes through the focus is called the axis, and the point in which it cuts the curve is called the principal vertex.

3 3

Thus: draw N,P,W,, N,P,W,, Nr N,P,W,, KASX, through the points N, P1, P2, P3, S, perpendicular to the directrix; each of these lines is a diameter; P,, P2, P3, A, are the ver- K tices of these diameters; ASX is the axis of the parabola, A the principal

vertex.

N3

P3

W3

6. A straight line which meets the ข curve in any point, but which, when produced both ways, does not cut it, is called a tangent to the curve at that point.

7. A straight line drawn from any point in the curve, parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter.

8. The ordinate which passes through the focus, is called the parameter of that diameter.

9. The part of a diameter intercepted between its vertex and the point in which it is intersected by one of its own ordinates, is called the abscissa of the diameter.

10. The part of a diameter intercepted between one of its own ordinates and its intersection with a tangent, at the extremity of the ordinate, is called the sub-tangent of the diameter. Thus let TPt be the tangent at P, the vertex of the diameter PW.

From any point Q in the curve, draw Qq parallel to Tt and cutting PW in v. Through S draw RS parallel to Tt.

Let QZ, a tangent at Q, cut WP, produced in Z.

Z

Then Qq is an ordinate to the diameter PW; Rr is the parameter of PW.

W

Pv is the abscissa of PW, corresponding to the point Q. vZ is the sub-tangent of PW, corresponding to the point Q. 11. A stright line drawn from any point in the curve, perpendicular to the axis, and terminated both ways by the curve, is called an ordinate to the axis.

12. The ordinate to the axis which passes through the focus is called the principal parameter, or latus rectum of the parabola.

13. The part of the axis intercepted between its vertex and the point in which it is intersected by one of its ordinates, is called the abscissa of the axis.

14. The part of the axis intercepted between one of its own

ordinates, and its intersection with a tangent at the extremity of the ordinate, is called the sub-tangent of the axis.

Thus from any point P in the curve draw Pp perpendicular to AX and cutting AX in M. Through S draw LSI perpendicular to AX.

Let PT, a tangent at P, cut XA Tproduced in T.

Then, Pp is an ordinate to the axis; Ll is the latus rectum of the curve.

AM is the abscissa of the axis corresponding to the point P.

A S

MX

MT is the subtangent of the axis corresponding to the point P.

It will be proved in Prop. III, that the tangent at the principal vertix is perpendicular to the axis; hence, the four last definitions are in reality included in the four which immediately precede them.

Cor. It is manifest from Def. 1, that the parts of the curve on each side of the axis are similar and equal, and that every ordinate Pp is bisected by the axis.

15. If a tangent be drawn at any point, and a straight line be drawn from the point of contact perpendicular to it, and terminated by the curve, that straight line is called a normal.

16. The part of the axis intercepted between the intersections of the normal and the ordinate, is called the sub-normal. Thus: Let TP be a tangent at any

point P.

From P draw PG perpendicular to the tangent, and PM perpendicular to the axis.

Then PG is the normal corresponding to the point P; MG is the sub-normal corresponding to the point P.

T

P

S M