.. KR2 : h'n'. r'n' in a constant ratio,

but r'n' = KN cos 0, where is the angle between KN and pq, and is constant;

.·. KR2= KN multiplied by some constant, or the locus of K is a parabola.

Case II. Let the plane of section meet all the generating lines on the same side of the vertex (Fig. 141).

Let a'v'b' be the elevation of the cone, v the plan of the vertex, and ab the diameter of the circular base parallel to the

ground line and therefore the plan of a'b'. Let the plane of section be perpendicular to the vertical plane of projection, and

draw its horizontal trace Im perpendicular to xy and its vertical trace cutting a'v' in h', and b'v' in k'. Project h' to hon av and k' to k on bv, then h and k are the plans of the points in which the generators through a and b meet the section plane, i.e. are the plans of two points on the required curve of intersection.

Imagine the cone cut by any horizontal plane as p'q' between h' and k', the elevation of the curve of intersection will be the line p'q', meeting a'v' in p' and b'v' in q' and lk' in r'; and the plan will be the circle on pq as diameter, obtained by projecting p'on av and q' on bv. The required plane of section cuts this plane of circular section in a line the elevation of which is r', and the plan of which is rr, projected from r'. If rr, meets the circle on pq in the points r and r,, these are the plans of two points of the required curve of intersection. Similarly the plans of any additional number of points can be obtained.

Rotate the plane of section round its horizontal trace till it coincides with the horizontal plane of projection; in elevation h', r' and k' travel to H', R', and K', and on plan h, r, r, and k travel along hH, rR, r ̧R ̧, kK, perpendicular to lm till they meet the projections of H', R' and K'. The points H, R, K, R, are points on the real outline of the required curve of intersection. It is an ellipse having HK as a diameter, and RR, as corresponding double ordinate.

Case III. Let the section plane cut both sheets of the cone (Fig. 142).

Let a'v'b' be the elevation of the cone, v the plan of v' the vertex, and ab the diameter of the circular base parallel to the ground line, and therefore the plan of a'b'. Let the plane of section be perpendicular to the vertical plane of projection, and draw its horizontal trace Im perpendicular to xy, and its vertical trace l' cutting b'v' in l', and a'v' in k'. Project h' to h on bv, and k' to k on av, then h and k are the plans of the points in

which the generators through a and b meet the section plane, i.e. are the plans of two points on the required curve of intersec

tion. Imagine the cone cut by any horizontal plane as p'q'; the elevation of the circle in which this plane meets the cone will be

the line p'q' meeting a'v' in p', b'v' in q', and lk' in r', and the plan will be the circle on pq as diameter obtained by projecting p' and q on av and be respectively. The required plane of section cuts this plane of circular section in a line, the elevation of which is r', and the plan of which is rr, projected from r'. If rr, meets the circle on pq in the points and r,, these are the plans of two points of the required curve of intersection. Similarly the plans of any additional number of points can be obtained.

Rotate the plane of section round its horizontal trace lm till

it coincides with the horizontal plane of projection; in elevation h', and ' travel to H', R' and K', and on plan h, r, r, and k travel along hH, rR, r ̧R1, kK perpendicular to lm till they meet the projections of H', R' and K'. The points H, R, R1 and K are points on the real outline of the required curve of intersection. It is an hyperbola having HK as a diameter, and RR, as corresponding double ordinate of the branch through H.

1

The asymptotes are parallel to the generators of the cone which are parallel to the plane of section. If therefore v'w' be drawn parallel to meeting xy in w', and w' be projected to meet the circular base ab in w and w1, the plans of the asymptotes

1

will be parallel to vw, vw,· Bisect hk in c, and draw cW, CW, parallel respectively to vw and vw,, and meeting Im in W, W, which will be points on the asymptotes, and they can therefore be drawn through C the point of bisection of HK.

EXAMPLES ON CHAPTER IX.

1. AVA,, an isosceles triangle, obtuse angled at V, is the elevation of a cone. Shew that if VB be drawn meeting AA, in

2

=

B, and such that VB AB. BA, (Ex. 15, Chap. 11.) and any plane be drawn having its vertical trace parallel to VB, and horizontal trace perpendicular to AA,, it will cut the cone in a rectangular hyperbola.

2. Given a cone and a point inside it determine the conics which have the given point as focus.

[Draw an elevation a''b' on a plane parallel to the plane containing the axis of the cone and the given point, and let f' be the elevation of the given point. The vertical traces of the required planes of section must be tangents at f' to the circles touching av and b'v', and passing through f'. Two solutions are generally possible.]

3. Shew that all sections of a right cone, made by planes parallel to tangent planes of the cone, are parabolas, and that the foci lie on a cone having with the first a common vertex and axis.

[Shew that the foci of parallel sections lie on a straight line through the vertex.]

4. Find the least angle of a cone from which it is possible to cut an hyperbola, whose eccentricity shall be the ratio of two to one.

5. Cut from a right cylinder an ellipse whose eccentricity shall be the ratio of the side of a square to its diagonal.

[In the cylinder inscribe a sphere, centre C; determine a point X in the horizontal plane through the centre such that

r = the above ratio, where r is the radius of the sphere. The

CX

required plane of section must be a tangent plane to the sphere through the point X.]

6. Shew how to cut from a given cone a hyperbola whose asymptotes shall contain the greatest possible angle.

[The plane of section must be parallel to the axis, pp. 246 and 241.]

7. Cut from a given cone the hyperbola of greatest eccentricity.

[The plane of section must be parallel to the axis, p. 248.]

8. Different elliptic sections of a right cone are taken having equal major axes; shew that the locus of the centres of the sections is a spheroid, oblate or prolate, according as the vertical angle of the cone is greater or less than 90°.

[Consider a series of sections perpendicular to a principal section of the cone. The centre is a fixed point on a line of constant length (the major axis), sliding between two fixed lines (the two generators of that section). It therefore traces out an ellipse which by revolution round the axis of the cone generates a spheroid.]