which, being the third equation, proves that the same value of ρ. satisfies it also. The three planes consequently all pass through the same straight line.

Ex. 7.

To find the locus of a point, the sum of the squares of whose distances from a number of given points has a given value.

Let p denote the sought point; a, ẞ,... the given ones; then

(p − a)2 + (p − ẞ)2 + &c. = Σ (p − a)2 = -C.

This is the equation of a sphere, the vector to whose centre is

i.e. the centre of inertia of the n points taken as equal.

and

Transpose the origin to this point, then (36)

Σ. α = 0,

p2 = − = {≥ (a2) + C' }.

Hence, that there may be a real locus, C must be positive and not less than the sum of the squares of the distances of the given system of points from their centre of inertia. If C have its least value, we have of course

p3 0,

the sphere having shrunk to a point.

ADDITIONAL EXAMPLES TO CHAP. V.

1. If two circles cut one another, and from one of the points of section diameters be drawn to both circles, their other extremities and the other point of section will be in a straight line.

2. If a chord be drawn parallel to the diameter of a circle, the radii to the points where it meets the circle make equal angles with the diameter.

3. The locus of a point from which two unequal circles subtend equal angles is a circle.

4. A line moves so that the sum of the perpendiculars on it from two given points in its plane is constant. Shew that the locus of the middle point between the feet of the perpendiculars is a circle.

5. If 0, O' be the centres of two circles, the circumference of the latter of which passes through 0; then the point of intersection A of the circles being joined with O' and produced to meet the circles in C, D, we shall have

AC.AD=2A0o.

6. If two circles touch one another in O, and two common chords be drawn through O at right angles to one another, the sum of their squares is equal to the square of the sum of the diameters of the circles.

7. A, B, C are three points in the circumference of a circle; prove that if tangents at B and C meet in D, those at C and A in E, and those at A and B in F; then AD, BE, CF will meet in a point.

8. If A, B, C are three points in the circumference of a circle, prove that V(AB. BC. CA) is a vector parallel to the tangent at A.

9.

A straight line is drawn from a given point 0 to a point P on a given sphere: a point Q is taken in OP so that

OP. OQ = k3.

Prove that the locus of Q is a sphere.

235

10. A point moves so that the ratio of its distances from two given points is constant. Prove that its locus is either a plane or a sphere.

11. A point moves so that the sum of the squares of its distances from a number of given points is constant.

its locus is a sphere.

Prove that

12. A sphere touches each of two given straight lines which do not meet; find the locus of its centre.

CHAPTER VI.

THE ELLIPSE.

43. 1. 1. IF we define a conic section as "the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line" (Todhunter, Art. 123), we shall find the equation to be (Ex. 5, Art. 35)

When e is less than 1, the curve is the ellipse, a few of whose properties we are about to exhibit.

2. SA, SA' are multiples of a: call one of them xa: then, by equation (1), putting xa for p, we get

the major axis of the ellipse, which we shall as usual abbreviate

and if vector CS be designated by a', CP by p', we have

whence, by substituting in (1), the equation assumes the form

212

`a3p'2 + (S'a'p')2 = − a1 (1 − e3) ;

which we may now write, CS being a and CP P,

3. This equation might have been obtained at once by referring the ellipse to the two foci, as Newton does in the Principia, Book 1. Prop. 11; the definition then becomes

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