v'os represent the projection of the triangle, v'm and P'n perpendicular to os being the projections of vw and Pp. Let v'm=h, om=a, om'=ß; also let on=x, on'=y, P'n, Osx' and ot=y'; then

In like manner, imagining the triangle ovo to be projected

Substituting these values of x and y' in the equation

2
x'2+y'2 = r2

for the circular base of the cone, the latter becomes

(hx−az)2+(hy—ẞz)2 = r2 (h—z)2,

which is the equation for any point in the surface of the

cone.

Or, if ov be supposed to lie in the plane zoY, VQ passing through P in any other plane, as MVQ; then a=0, and the equation becomes

h2x2 + (hy — ẞz)2 = r2 (h—z)2.

(a)

When the cone is upright, vw coincides in direction with oz; and since then a=0, B=0, we should have

p2
h2

h2x2 + h2 y2 = r2 (h—x)2 or x2+y2 = (h−z)2 (b)

for the equation of an upright cone having a circular base.

For an upright cone having an elliptical base: -Let the origin of the co-ordinates be at the centre, and let ox, or coincide with the semi-transverse and semi-conjugate axes (t and c): also let the co-ordinates of P and of Q be represented as before. Then a and 6 being each zero, the above values of x and y become

substituting these values in t2y'2+c2x2=c2, the equation [(6) Art. 56.] for the elliptical base, the latter becomes,

t2 c2

h2

t2 h2x2 + c2 h2y2=t2 c2 (h—z)2 or t2x2+c2y2 = -(h—z)2, (c) which is the equation for such a cone.

If the origin of the co-ordinates be at v the vertex of an upright cone, r being the radius of the base if circular, or t and e the semi-axes of the base if elliptical; and h being the height, as before; then, z being measured from v, on putting z for h-x in the second members of (b) and (c), the equation for the upright cone will become, when the base is circular,

185. To find the equation for the curve line formed by the intersection of a plane with the convex surface of an upright cone having a circular base.

Z

Let the cutting plane pass through ox, and be inclined to the base of the cone in an angle expressed by 0. Then, if P be a point in the curve line of intersection, its co-ordinates with respect to zox, ZOY, XOY may, as in Art. 180., be represented by x, y, z, and with respect to XOY' by ' and y', z' being zero; therefore we shall have

x=x', y=y' cos. 6, and z=y' sin. 0. These being substituted in the equa

tion

/2

h2

(Art. 184.)

or h2x2+y2 (h2 cos.2 0-2 sin.2 0)+2r2 hy' sin. =r2 h2;

which is therefore an equation for the curve line.

If the coefficient of y'2 is positive, the curve (Art. 48. a.) is an ellipse; if negative (Art. 48. c.), an hyperbola; and if zero (Arts. 46. and 48. e.), a parabola. Hence, if 2 cos.20

h

r

is greater than r2 sin.2 0, or tan. is less than; that is, if 0 is less than the angle made by the side of the cone with the base, the section is an ellipse; if is greater than such angle, the section is an hyperbola; and if equal to it, the section is a parabola.

a. If an oblique cone have a circular base, and a section be made through ox (Fig. to Art. 185.); on substituting the above values of x, y, z in the equation (a) Art. 184., there would be obtained for the section

Y

an equation from which the section might be proved to be that of a circle when, as in the annexed figure, the angle YAO, which the cutting plane makes with VY, is equal to the supplement of the angle VY'O (the plane VY'Y being perpendicular to the base), which the base makes with the opposite side of the cone; but this may be more easily proved by the Elements of Geometry. For, let VY' be produced till it meets AO produced in B; then the triangles YOA, Y'OB will be similar to one another; whence

Χ

YO: OA:: OB: OY', and OY. OY'=OA. OB.

B

But the base YXY' being a circle, OY.OY' (Euc. 35. III.)=0x2; therefore OA. OB=OX2, which is a property of a circle; and consequently the section AXBX' is a circle.

186. DEF. 21. A section formed in an oblique cone by a plane which makes the same angle with one side of the cone which the plane of the base makes with the opposite side, in a plane passing through the axis, is called a subcontrary section of the cone.

PROPOSITION VIII.

187. To find the equations for a spheroid, an hyperboloid, and a paraboloid.

Let the co-ordinate axes be rectangular, and let zox be part of an ellipse of which o is the centre; also, let, for example, ox be the semi-transverse (t), and oz the semi

conjugate axis (=c); then, if any point, as Q, in the periphery

revolve about oz, it will evidently describe the circumference of a circle QR, whose centre is at A, where a perpendicular to oz from Q meets that line. Here QR is a variable circle; and if its radius be represented by r, then r must be understood to vary with the place of that circle.

Let ZPM be any position of the revolving ellipse, and let the co-ordinates of P with respect to ZOX, ZOY, XOY be x, y, z; also, let the co-ordinates of P with

Н

K

M'

R

M

K

respect to zoм be z and r (=OA and AP); then the equation for P in the circle QR will be (Art. 56. b.)

x2+ y2=r2;

and in the ellipse [(a) Art. 56.]

t2

(c2-22)=r2, or t2 c2=c2 r2 +t2z2.

c2

Substituting the first value of 2 in the last equation, the latter becomes

t2 c2=c2 (x2+y2)+t2 z2,

which is the equation for any point P in the surface of the spheroid.

The equation for an hyperboloid, of which figure suppose H'ZK' to be a portion, is found in like manner, by substituting the first value of 2 in the equation

for the hyperbola ZP'M'; from which we obtain for the required equation

· t2x2 — t2 c2 = ç2 (x2+y2).

The equation for a paraboloid, of which figure suppose HZK to be a portion, is found by substituting the first value of 2 in the equation pz'=r2 (Art. 72.) for the parabola ZPM, p being put for the parameter of the axis zo, and ≈' for Thus we obtain

ᏃᎪ.

px' = x2 + y2

for the equation of the paraboloid.

CHAP. VI.

CURVE SURFACES OF THE SECOND ORDER WHICH ARE NOT PRODUCED BY REVOLUTION.

188. THE most general equation for such surfaces is

Ax2 + Bу2 + cz2 + DXY + EXZ + FYZ + GX + Hy + KZ = L, the co-ordinate axes being rectangular or oblique.

(A)

189. If the equations of any straight line intersecting a curve surface represented by the above equation be

x=az+b, y=a'z+b',

on substituting the second members in place of x and y in that equation, the result will be a quadratic equation with respect to the variable quantity z; therefore there can be but two real values of that variable, and consequently also but two real values of x and y for the points in which the straight line intersects the curve. Thus a straight line can cut a curve surface of the second order in only two points. Such line is called a chord of the curve surface.

PROPOSITION I.

190. If a plane bisect any three parallel chords not in one plane, it will bisect every chord parallel to these.

Let the bisected portions of the three given chords be represented by +k' and -k', +k" and ―k", +k'"' and—k'"', and let the co-ordinates of the middle points be (y', d', '), (y'', d'', {''), (y''', ''', '''); then the co-ordinates of the extremities of +k', +k", +k" in the curve surface may be represented by

in which u, v, t denote the ratios between the sines of the angles which k', k", k'", and lines imagined to be drawn from their extremities and the extremities of their co-ordinates, with respect to the middle points, make with one another