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

[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 show that

is an invariant. This may most easily be done by proving that

Perhaps, however, it is simpler to write a for V.ẞy, and we thus obtain

[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

Spop=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.

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 plane sections of an ellipsoid, erected at the centre, their lengths being the principal semi-axes of the sections. [Fresnel's Wave-Surface.]

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 diameters.

13. Show 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 show 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

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.

19. The sum of the reciprocals of the squares of the perpendiculars from the centre upon three conjugate tangent planes is constant.

C C

20. Cones are drawn, touching an ellipsoid, from any two points of a similar, similarly situated, and concentric ellipsoid. Show that they intersect in two plane curves.

Find the locus of the vertices of the cones when these plane sections are at right angles to one another.

21. Find the locus of the points of contact of tangent planes which are equidistant from the centre of a surface of the second order.

22. From a fixed point A, on the surface of a given sphere, draw any chord AD; let D' be the second point of intersection of the sphere with the secant BD drawn from any point B; and take a radius vector AE, equal in length to BD', and in direction either coincident with, or opposite to, the chord AD: the locus of E is an ellipsoid, whose centre is 4, and which passes through B. (Hamilton, Elements, p. 227.)

23. Show that the equation

l2 (e2 — 1) (e + Saa') =

(Sap)2-2e Sap Sa'p+(Sa'p)2 +(1—e2)p2,

where e is a variable (scalar) parameter, represents a system of confocal surfaces. (Ibid. p. 644.)

24. Show that the locus of the diameters of

Spøp = 1

which are parallel to the chords bisected by the tangent planes to the cone

is the cone

Spyp = 0

5.ρφψ' φρ = 0.

25. Find the equation of a cone, whose vertex is one summit of a given tetrahedron, and which passes through the circle circumscribing the opposite side.

26. Show that the locus of points on the surface

Spop = 1,

the normals at which meet that drawn at the point pw, is on the cone

S.(p-w) pwop = 0.

27. Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane.

28. Show that the locus of the pole of the plane

is a sphere, if a be subject to the condition

Sapa = 0.

29. Show that the equation of the surface generated by lines drawn through the origin parallel to the normals to

30. Common tangent planes are drawn to

2 Sλp Sup+(p-Sλμ)p 1, and Tp

=

find the value of h that the lines of contact with the former surface may be plane curves. What are they, in this case, on the sphere?