If it is desired that the hyperboloid which has become a straight line should be a cylinder of definite diameter, then another surface of revolution will have to take the place of the other hyperboloid. It will now be shown how this other surface of revolution may be determined.

Referring to Fig. 646, bob is the plan and b'o'b' is the elevation of the axis of the cylinder which is assumed to be horizontal.

plan and o' is the elevation of the axis of the other surface or solid which is assumed to be horizontal and perpendicular to the vertical plane of projection. This other solid will be called the roll. o is the plan and o,'o'o' is the elevation of the common perpendicular to the two given axes. Making o'o' equal to the radius of the cylinder determines o'o' the radius of the roll at the throat. The plane of the throat circle of the roll intersects the cylinder in an ellipse which touches the throat circle at oo'. Any other plane parallel to the plane of the throat circle will intersect the cylinder in an ellipse and the roll in a circle, and the ellipse and circle will touch one another. Moreover all such sections of the cylinder will be exactly alike.

HT is the horizontal trace of a plane parallel to the plane of the throat circle of the roll. This plane intersects the cylinder in an ellipse of which m'n' is the elevation. A circle with

aoa is the

id

b

o as centre drawn to touch the ellipse m'n' is the elevation of the circular section of the roll by the plane HT, and this determines the points e and ƒ on the horizontal meridian section of the roll. A projector from s', the point of contact of the ellipse and circle, to HT determines 8 a point on the plan uov of the curve of contact between the cylinder and roll.

The radius of the circular section of the roll at HT and the point of contact 88' may be found without drawing the ellipse m'n'. It is evident that if the ellipse m'n' be moved to the right a distance equal to w on the plan it will then coincide with the ellipse c'd' which is the elevation of the section of the cylinder by the plane of the throat circle of the roll. And if the circle e'f' be moved an equal distance to

the right it will then be in the position ef and will touch the ellipse c'd' at s'. Hence the circle efi may be drawn first and then the circle e'f' which is equal to it. Projecting from 8' to 8, and making 8,8 equal to w determines the point s. In like manner, by taking other positions for the plane HT any number of points on the horizontal meridian section of the roll and any number of points on the plan uov of the curve of contact may be determined. The ellipse c'd' is therefore the only ellipse which need be drawn but it should be constructed as accurately as possible, to ensure a good result.

298. The Paraboloid of Revolution. The paraboloid of revolution may be generated by the revolution of a parabola about its axis. Sections of the surface by planes parallel to or containing the axis are equal parabolas. Sections by planes perpendicular to the axis are circles. All other plane sections are ellipses.

299. The Ellipsoid.-The ellipsoid may be generated by a variable ellipse which moves so that its plane is parallel to a fixed plane and the extremities of its axes are on two fixed ellipses which have one common axis and which have their planes at right angles to one another and to the fixed plane.

Referring to Fig. 647, let the horizontal plane be the fixed plane. Let oa, o'a' and ob, o'b' be the semi-axes of one fixed ellipse whose plane is horizontal, and let ob, o'b

and oc, o'c' be the semiaxes of the other fixed ellipse whose plane is parallel to the vertical plane of projection. A moving variable ellipse whose axes are horizontal chords of these two fixed ellipses generate an ellipsoid.

The plan of the horizontal fixed ellipse is the plan of the ellipsoid, and the elevation of the other fixed ellipse which is parallel to the vertical plane of projection is an elevation of the ellipsoid. def is

the half plan and d'e'f' is the elevation of the moving variable ellipse in one position. oe on the half plan is equal to o'e' on the

part elevation (v) which is a projection on a vertical plane at right angles to the plane of the elevation (u).

All plane sections of the ellipsoid are either ellipses or circles.

If go'd', a diameter of the ellipse which is the elevation of the ellipsoid, be taken as the vertical trace of a plane which is perpendicular

to the vertical plane of projection, the elliptic section of the ellipsoid by this plane will have one semi-axis equal to o'g' and the other semi-axis equal to oa. If o'g' is equal to oa then the section is evidently a circle. This suggests the construction for finding the plane containing the centre of the ellipsoid and cutting the ellipsoid in a circle. There are evidently two such planes.

All plane sections parallel to one circular section are circles. In Fig. 647, g'o'd' is the elevation of one circular section of the ellipsoid and the chords of the ellipse parallel to g'o'd' are the elevations of other circular sections. The elevations of the centres of these circles lie on the diameter m'o'n' which is conjugate to the diameter g'o'd'.

As the plane of a circular section moves away from the centre of the ellipsoid the circle gets smaller and smaller until the plane becomes tangential to the ellipsoid at mm' or nn' when the circle becomes a point which is called an umbilic. There are evidently four umbilics on an ellipsoid.

At (w) in Fig. 647 is shown a projection of the ellipsoid on a plane parallel to planes of circular sections. On this projection nine circular sections are shown.

The curve of contact between an ellipsoid and an enveloping cylinder or an enveloping cone is a plane section of the ellipsoid. The outline of the projection (w) in Fig. 647 is the trace of a cylinder which envelops the ellipsoid and is perpendicular to the plane of the projection; the diameter i'o'' of the ellipse (u) is the elevation of the curve of contact.

The tangent plane to the ellipsoid at any point on it contains the tangents at that point to any two plane sections of the ellipsoid through the point. When two of the three axes of an ellipsoid are equal it becomes a spheroid.

300. The Hyperboloid of One Sheet.-The hyperboloid of one sheet may be generated by a variable ellipse which moves so that its plane is always parallel to a fixed plane and has the extremities of its axes on two fixed hyperbolas whose planes are perpendicular to one another and to the fixed plane and which have a common conjugate axis.

Referring to Fig. 648, OA and OB are the semi-transverse axes of two fixed hyperbolas and OC is a common semi-conjugate axis. OA and OB are horizontal and OC is consequently vertical. OA is parallel to the plane of the elevation (v). The moving variable ellipse is in this case always horizontal, and it will be smallest when its plane coincides with AOB, and it is then the throat ellipse or principal elliptic section of the hyperboloid. In this article when "hyperboloid" is mentioned "hyperboloid of one sheet" will be understood.

The moving ellipse remains similar to the throat ellipse, hence if DD and EE are the axes of the moving ellipse in one position de is parallel to ab.

At (w) is shown an elevation of the hyperboloid on a plane

parallel to OB. The true form of one fixed hyperbola is shown in the elevation (v) and the true form of the other is shown in the elevation (w).

The asymptotic cone of the hyperboloid is generated by a variable ellipse which moves so that its plane is parallel to the throat ellipse and has the extremities of its axes on the asymptotes of the fixed hyperbolas. The vertex of this cone is at O. GG and HH are the axes of this moving variable ellipse in one position. This generating ellipse is similar to the throat ellipse, hence gh is parallel to ab. The ellipse which is the section of the asymptotic cone by a plane parallel

to the plane of the throat ellipse and at a distance from it equal to OC is evidently equal to the throat ellipse.

Sections of the hyperboloid and of the asymptotic cone by the same or by parallel planes are curves of the same kind.

If the transverse axes of the fixed hyperbolas are equal the hyperboloid becomes an hyperboloid of revolution.

The plane of a circular section of the hyperboloid may be found as follows. Let OA be greater than OB. Then on the elevation (w) take o' as centre and with a radius equal to oa describe an arc to cut the hyperbola b'k'e' at k'; then k'o'k' is the vertical trace of a plane, perpendicular to the plane of the elevation (w), which will cut the

hyperboloid in a circle whose radius is equal to o'k'. All sections by planes parallel to this plane will be circles. There are evidently two systems of parallel planes which will cut the hyperboloid in circles.

The hyperboloid of one sheet may also be generated by a moving variable hyperbola having a fixed conjugate axis, and having the extremities of its transverse axis on a fixed ellipse whose plane bisects at right angles the fixed conjugate axis of the moving hyperbola. The fixed ellipse is the throat ellipse of the hyperboloid.

Referring to the plan (u) and elevation (v) Fig. 648, the straight lines RSR and TST lie on the hyperboloid. These lines are in a vertical plane and their plans coincide and are tangential to the plan of the throat ellipse. The hyperboloid may therefore be generated by the motion of one or other of these straight lines. These two lines belong to two different systems. A line of one system will never meet any other line of that system but it will intersect every line of the other system if the lines are produced far enough. Hence if a straight line moves in contact with any three lines of one of the systems it will generate the hyperboloid.

301. The Hyperboloid of Two Sheets.-The hyperboloid of two sheets may be generated by a variable ellipse which moves so that its plane is always parallel to a fixed plane and has the extremities of its axes on two fixed hyperbolas whose planes are perpendicular to one another and to the fixed plane and. which have a common transverse axis.

Referring to Fig. 649, which is a pictorial projection of one quarter of an hyperboloid of two sheets, OB and OC are the semi-conjugate axes of two fixed hyperbolas and OA is a common transverse axis. OA and OB are horizontal and OC is vertical. The plane containing OB and OC is the fixed plane referred to in the above definition. ADE is part of one branch of one fixed hyperbola having OA for its semi-transverse axis and OC for its semi-conjugate axis; the plane of this hyperbola is vertical. AFH is part of one branch of the other fixed hyperbola having OA for its semi-transverse axis and OB for its semi