Again, if a, ẞ, y be any rectangular unit-vectors

And so on. These elementary investigations are given here for the benefit of those who have not read Chapter V. The student may easily obtain all such results in a far more simple manner by means of the formulae of that Chapter.

265.] Find the locus of intersection of a rectangular system of three tangents to an ellipsoid.

If be the vector of the point of intersection, a, ß, y the tangents, then, since + xa should give equal values of a when substituted in the equation of the surface, giving

or

we have

S(+xa) & ( + xa) = 1,

x2Sapa + 2xSpa+ (Swpw−1) = 0,
(Sopa)2 = Sapa (Swpw—1).

Adding this to the two similar equations in ß and y

(Sapw)2 + (SBpw)2 + (Sy¢w)2

=

(Sapa+SBo3+ Sypy) (Swów -1),

266.] If a rectangular system of chords be drawn through any point within an ellipsoid, the sum of the reciprocals of the rectangles under the segments into which they are divided is constant.

With the notation of the solution of the preceding problem, w giving the intersection of the vectors, it is evident that the product of the values of x is one of the rectangles in question taken negatively.

This evidently depends on Swow only and not on the particular directions of a, ẞ, y: and is therefore unaltered if be the vector of any point of an ellipsoid similar, and similarly situated, to the given one. [The expression is interpretable even if the point be exterior to the ellipsoid.]

267.] Shew that if any rectangular system of three vectors be drawn from a point of an ellipsoid, the plane containing their other extremities passes through a fixed point. Find the locus of the latter point as the former varies.

With the same notation as before, we have

and those of two other vectors similarly determined. It therefore passes through the point whose vector is

Thus the first part of the proposition is proved.

the equation of a concentric ellipsoid.

268.] Find the directions of the three vectors which are parallel to a set of conjugate diameters in each of two central surfaces of the second degree.

be

Transferring the centres of both to the origin, let their equations

If a, 3, y be vectors in the required directions, we must have (§ 254)

(2)

From these equations pa ||By | ya, &c.

Hence the three required directions are the roots of

This is evident on other grounds, for it means that if one of the surfaces expand or contract uniformly till it meets the other, it will touch it successively at points on the three sought vectors.

We may put (3) in either of the following forms

and, as and are given functions, we find the solutions by the processes of Chapter V.

[Note. As 4-1 and -1 are not, in general, self-conjugate functions, equations (4) do not signify that a, ß, y are vectors parallel to the principal axes of the surfaces

-1

In these equations it does not matter whether 4-1y is self-conjugate or not; but it does most particularly matter when they are differentiated, so as to find axes, &c.]

Given two surfaces of the second degree, there exists in general a set of Cartesian axes, whose directions are those of conjugate diameters in every one of the surfaces of the second degree passing through the intersection of the two surfaces given.

L

For any surface through the intersection of

=

Sppp 1 and S(p-a)(p-a) = e, is fSpop-S(pa) (p-a) = ƒ—e,

where f and e are scalars.

The axes of this depend only on the term

Sp (fo-4)p.

Hence the set of conjugate diameters which are the same in all are the roots of

V(fø—¥)p(£14-) p = 0, or Vop¥p = 0,

as we might have seen without analysis.

The locus of the centres is given by the equation

(4-ƒ)p—a = 0,

where f is a scalar variable.

269.] Find the equation of the ellipsoid of which three conjugate semi-diameters are given.

Let the vector semi-diameters be a, ß, y, and let

and similarly for the other combinations. Thus, as we have

we find at once

ρδ.αβγ = αδ.βγρ + βδ.γαρ + γδ.αβρ,

φρδ.αβγ = βγδ.βγρ + Γγαδ.γαρ + Γαβδ.αβρ ;

and the required equation may be put in the form

82.aẞy = 82.aẞp+S2.Byp+S2.yap.

The immediate interpretation is that if four tetrahedra be formed by grouping, three and three, a set of semi-conjugate vector axes of an ellipsoid and any other vector of the surface, the sum of the squares of the volumes of three of these tetrahedra is equal to the square of the volume of the fourth.

270.] When the equation of a surface of the second order can be put in the form Spo-1p = 1, where

(p −g) (p −91) ($—92) = 0,

(1)

we know that 9,91, 92 are the squares of the principal semi-diameters. Hence, if we put 4+h for 4 we have a second surface, the differences of the squares of whose principal semiaxes are the same as for the first. That is, Sp(p+ h)−1p = 1 ........ (2)

is a surface confocal with (1). From this simple modification of the equation all the properties of a series of confocal surfaces may easily be deduced. We give one as an example.

271.] Any two confocal surfaces of the second order, which meet, intersect at right angles.

For the normal to (2) is, evidently,

and that to another of the series, if it passes through the common. point whose vector is P, is there

But

(p + h1)−1p.

1

S.(p+ h)−1p(p+h2)−1p = S.o (p + h) ($ + hy) o

=
=

1

h-hi

Sp ((p + h1)−1 — ($+ h)−1) p,

and this evidently vanishes if h and h1 are different, as they must be unless the surfaces are identical.

272.] To find the conditions of similarity of two central surfaces of the second order.

Referring them to their centres, let their equations be

=

Spopp 1,1
Spo'p 1.S

=

Now the obvious conditions are that the axes of the one are proportional to those of the other. Hence, if

be the equations for determining the squares of the reciprocals of the semiaxes, we must have

but two scalar conditions necessary. Eliminating μ we have