We see at once that any two of these equations give the third by addition or subtraction, which is the proof of the theorem.

214.] Given any number of points A, B, C, &c., whose vectors (from the origin) are a1, a2, az, &c., find the plane through the origin for which the sum of the squares of the perpendiculars let fall upon it from these points is a maximum or minimum.

be the required equation, with the condition (evidently allowable)

To 1.

The perpendiculars are (§ 208) -1Swa1, &c.

and the condition that is a unit-vector gives

Sada = 0.

Hence, as da may have any of an infinite number of values, these equations cannot be consistent unless

where x is a scalar.

Σ.αδαπ = 2π,

The values of a are known, so that if we put

Σ.αδαπ = φω,

o is a given self-conjugate linear and vector function, and therefore x has three values (91, 92, 93, § 164) which correspond to three mutually perpendicular values of w. For one of these there is a maximum, for another a minimum, for the third a maximumminimum, in the most general case when 91, 92, 93 are all different.

215.] The following beautiful problem is due to Maccullagh. Of a system of three rectangular vectors, passing through the origin, two lie on given planes, find the locus of the third.

Let the rectangular vectors bew, p, σ. Then by the conditions of the problem Sap = Spo = Sow = 0, δαπ = SBp = 0.

and

The solution depends on the elimination of p and among these five equations. [This would, in general, be impossible, as p and w between them involve six unknown scalars; but, as the tensors are (by the very form of the equations) not involved, the five given equations are necessary and sufficient to eliminate the four unknown. scalars which are really involved. Formally to complete the requisite number of equations we might write To = a, Τρ but a and may have any values whatever. b

Tp = b,

the required equation. As will be seen in next Chapter, this is a cone of the second order whose circular sections are perpendicular to a and B. [The disappearance of x and y in the elimination instructively illustrates the note above.]

EXAMPLES TO CHAPTER VI.

1. What propositions of Euclid are proved by the mere form of the equation ρ (1−x) a + xß,

p =

which denotes the line joining any two points in space?

2. Shew that the chord of contact, of tangents to a parabola which meet at right angles, passes through a fixed point.

3. Prove the chief properties of the circle (as in Euclid, III) from the equation p = a cos 0 +ẞ sin 0 ;

where Ta Ts, and Saß = 0.

=

4. What locus is represented by the equation

where Ta = 1?

S2 ap + p2 = 0,

5. What is the condition that the lines

Vap = B, Γαρ = βι

intersect? If this is not satisfied, what is the shortest distance between them?

6. Find the equation of the plane which contains the two parallel lines Va (p-B) = 0, Va (p-ẞ1) = 0.

7. Find the equation of the plane which contains.

8. Find the equation of a straight line passing through a given point, and making a given angle with a given plane.

Hence form the general equation of a right cone.

S'p-315 = √ Sen 9. Frei

9. What conditions must be satisfied with regard to a number of given lines in space that it may be possible to draw through each of them a plane in such a way that these planes may intersect in a common line?

10. Find the equation of the locus of a point the sum of the squares of whose distances from a number of given planes is con

stant.

11. Substitute "lines" for "planes" in (10).

12. Find the equation of the plane which bisects, at right angles, the shortest distance between two given lines.

Find the locus of a point in this plane which is equidistant from the given lines.

13. Find the conditions that the simultaneous equations

where a, ß, y are any three vectors?

15. Find the equation of the plane which passes through two given points and makes a given angle with a given plane.

16. Find the area of the triangle whose corners have the vectors

a, ß, y.

Hence form the equation of a circular cylinder whose axis and radius are given.

17. (Hamilton, Bishop Law's Premium Ex., 1858).

(a.) Assign some of the transformations of the expression

where a and ẞ are the vectors of two given points A and B. (3.) The expression represents the vector y, or OC, of a point C in the straight line AB.

(c.) Assign the position of this point C.

18. (Ibid.)

(a.) If a, ß, y, d be the vectors of four points, A, B, C, D, what is the condition for those points being in one plane?

(b.) When these four vectors from one origin do not thus terminate upon one plane, what is the expression for the volume of the pyramid, of which the four points are the corners?

(c). Express the perpendicular & let fall from the origin O on the plane ABC, in terms of a, B, y.

19. Find the locus of a point equidistant from the three planes

20. If three mutually perpendicular vectors be drawn from a point to a plane, the sum of the reciprocals of the squares of their lengths is independent of their directions.

21. Find the general form of the equation of a plane from the condition (which is to be assumed as a definition) that any two planes intersect in a single straight line.

22. Prove that the sum of the vector areas of the faces of any polyhedron is zero.

CHAPTER VII.

THE SPHERE AND CYCLIC CONE.

216.] AFTER that of the plane the equations next in order of simplicity are those of the sphere, and of the cone of the second order. To these we devote a short Chapter as a valuable preparation for the study of surfaces of the second order in general.

217.] The equation Tp Ta,

or

Τρ

=

p2 = a2,

denotes that the length of p is the same as that of a given vector a, and therefore belongs to a sphere of radius Ta whose centre is the origin. In § 107 several transformations of this equation were obtained, some of which we will repeat here with their interpretations. Thus S(pa) (pa) = 0

shews that the chords drawn from any point on the sphere to the extremities of a diameter (whose vectors are a and a) are at right angles to each other.

T(pa) (pa) = 2TVap

shews that the rectangle under these chords is four times the area of the triangle two of whose sides are a and ρ.

p = (p+a)1a (p+ a) (see § 105)

shews that the angle at the centre in any circle is double that at the circumference standing on the same arc. All these are easy consequences of the processes already explained for the interpretation of quaternion expressions.

218.] If the centre of the sphere be at the extremity of a the equation may be written