273.] Find the greatest and least semi-diameters of a central plane section of an ellipsoid.

Here

Spopp 11

=

Sap = 0

(1)

together represent the elliptic section; and our additional condition

is that Tp is a maximum or minimum.

Differentiating the equations of the ellipse, we have

This shews that the maximum or minimum vector, the normal at its extremity, and the perpendicular to the plane of section, lie in one plane. It also shews that there are but two vector-directions which satisfy the conditions, and that they are perpendicular to each other, for (2) is satisfied if ap be substituted for p.

We have now to solve the three equations (1) and (2), to find the vectors of the two (four) points in which the ellipse (1) intersects the cone (2). We obtain at once

a quadratic equation in p2, from which the lengths of the maximum and minimum vectors are to be determined. By § 147 it may be mp1Sap ̄1a— p2S.a(m,—4) a+ a2 = 0.

written

....

(5)

[If we had operated by S.4-1a or by S.-1p, instead of by S.p, we should have obtained an equation apparently different from this, but easily reducible to it. To prove their identity is a good exercise for the student.]

Substituting the values of p2 given by (5) in (3) we obtain the vectors of the required diameters. [The student may easily prove directly that (1-pip)-1a and (1-p3)-1a

are necessarily perpendicular to each other, if both be perpendicular to a, and if på and på be different. See § 271.]

274.] By (5) of last section we see that

Also the locus of normals to all diametral sections of an ellipsoid, whose areas are equal, is the cone

When the roots of (5) are equal, i.e. when

(ma2-Sapa)2 = 4ma2Sap-1a,

(6)

the section is a circle. It is not difficult to prove that this equation is satisfied by only two values of Ua, but another quaternion form of the equation gives the solution of this and similar problems by inspection. (See § 275 below.)

275.] By § 168 we may write the equation

in the new form

8.ρμρ + χρ3 = 1,
Pp2

where p is a known scalar, and λ and μ are definitely known (with the exception of their tensors, whose product alone is given) in terms of the constants involved in 4. [The reader is referred again also to §§ 121, 122.] This may be written

28λρδμρ + (1-8λμ) ρα

= 1.

(1)

From this form it is obvious that the surface is cut by any plane perpendicular to λ or μ in a circle. For, if we put

we have

Sλp = a,
2aSup+(p-Sλu) p2 = 1,

the equation of a sphere which passes through the plane curve of intersection.

Hence λ and μ of § 168 are the values of a in equation (6) of the preceding section.

276.] Any two circular sections of a central surface of the second order, whose planes are not parallel, lie on a sphere.

For the equation (Sλp-a) (Sup-b) = 0,

where a and b are any scalar constants whatever, is that of a system of two non-parallel planes, cutting the surface in circles. Eliminating the product SλpSup between this and equation (1) of last section, there remains the equation of a sphere.

277.] To find the generating lines of a central surface of the second order.

then, if a be the vector of any point on the surface, and a vector parallel to a generating line, we must have

The first is the equation of a plane through the origin parallel to the tangent plane at the extremity of a, the second is the equation of the asymptotic cone. The generating lines are therefore parallel to the intersections of these two surfaces, as is well known.

where y is a scalar to be determined. Operating on this by S.ẞ and S.y, where ẞ and y are any two vectors not coplanar with a, we have Sw(yoẞ+ Vaß) = 0, St (yoy-Vya) = 0.

(1)

which, according to the sign of y, gives one or other generating line.

Here Vẞy may be any vector whatever, provided it is not perpendicular to a (a condition assumed in last section), and we may write for it 0.

Substituting the value of y before found, we have

or, as we may evidently write it,

= 4-1(V.a√ pa0) ± √ √μað.

Уфав.

(2)

m

[Any one of these sets of values forms the complete solution of the problem; but more than one have been given, on account of their singular nature and the many properties of surfaces of the second order which immediately follow from them. It will be excellent practice for the student to shew that

40=U(V.$aVap ̄10±√ =—=—= √pa0)

m

is an invariant. This may most easily be done by proving that V.4001 = 0 identically.]

Perhaps, however, it is simpler to write a for Vẞy, and we thus

obtain

zw=—¿ ̄1VaVapa√Vapa.

m

Тафа.

[The reader need hardly be reminded that we are dealing with the general equation of the central surfaces of the second order-the centre being origin.]

EXAMPLES TO CHAPTER VIII.

1. Find the locus of points on the surface

Spopp 1

where the generating lines are at right angles to one another.

2. Find the equation of the surface described by a straight line which revolves about an axis, which it does not meet, but with which it is rigidly connected.

3. Find the conditions that

Spopp=1

may be a surface of revolution, with axis parallel to a given vector. 4. Find the equations of the right cylinders which circumscribe a given ellipsoid.

5. Find the equation of the locus of the extremities of perpendiculars to central plane sections of an ellipsoid, erected at the

centre, their lengths being the principal semi-axes of the sections. [Fresnel's Wave-Surface. See Chap. XI.]

6. The cone touching central plane sections of an ellipsoid, which are of equal area, is asymptotic to a confocal hyperboloid.

7. Find the envelop of all non-central plane sections of an ellipsoid whose area is constant.

8. Find the locus of the intersection of three planes, perpendicular to each other, and touching, respectively, each of three confocal surfaces of the second order.

9. Find the locus of the foot of the perpendicular from the centre of an ellipsoid upon the plane passing through the extremities of a set of conjugate diameters.

10. Find the points in an ellipsoid where the inclination of the normal to the radius-vector is greatest.

11. If four similar and similarly situated surfaces of the second order intersect, the planes of intersection of each pair pass through a common point.

12. If a parallelepiped be inscribed in a central surface of the second degree its edges are parallel to a system of conjugate dia

meters.

13. Shew that there is an infinite number of sets of axes for which the Cartesian equation of an ellipsoid becomes

x2 + y2+z2 = e2.

14. Find the equation of the surface of the second order which circumscribes a given tetrahedron so that the tangent plane at each angular point is parallel to the opposite face; and shew that its centre is the mean point of the tetrahedron.

15. Two similar and similarly situated surfaces of the second order intersect in a plane curve, whose plane is conjugate to the vector joining their centres.

16. Find the locus of all points on

Sppp = 1,

where the normals meet the normal at a given point.

Also the locus of points on the surface, the normals at which meet a given line in space.

17. Normals drawn at points situated on a generating line are parallel to a fixed plane.

18. Find the envelop of the planes of contact of tangent planes drawn to an ellipsoid from points of a concentric sphere. Find the locus of the point from which the tangent planes are drawn if the envelop of the planes of contact is a sphere.