10. If a chord of a hyperbola be one diagonal of a parallelogram whose sides are parallel to the asymptotes, the other diagonal passes through the centre.

11. Show that

p = x2a + y2 ß + (x + y)2 y

is the equation of a cone of the second degree, and that its section by the plane

is an ellipse which touches, at their middle points, the sides of the triangle of whose corners a, ß, y are the vectors. (Hamilton, Elements, p. 96.)

12. The lines which divide, proportionally, the pairs of opposite sides of a gauche quadrilateral, are the generating lines of a hyperbolic paraboloid. (Ibid. p. 97.)

represents a cone of the third order, and that its section by the plane

are the asymptotes and the three (real) tangents of inflexion. Also that the mean point of the triangle formed by these lines is a conjugate point of the curve. Hence that the vector

a+B+y is a conjugate ray of the cone.

(Ibid. p. 96.)

45.

CHAPTER II.

PRODUCTS AND QUOTIENTS OF VECTORS.

WE

E now come to the consideration of points in which the Calculus of Quaternions differs entirely from any previous mathematical method; and here we shall get an idea of what a Quaternion is, and whence it derives its name. These points are fundamentally involved in the novel use of the symbols of multiplication and division. And the simplest introduction to the subject seems to be the consideration of the quotient, or ratio, of two vectors.

46. If the given vectors be parallel to each other, we have already seen (§ 22) that either may be expressed as a numerical multiple of the other; the multiplier being simply the ratio of their lengths, taken positively if they are similarly directed, negatively if they run opposite ways.

47. If they be not parallel, let ŌA and OB be drawn parallel and equal to them from any point 0; and the question is reduced to finding the value of the ratio of two vectors drawn from the same point. Let us try to find upon how many distinct numbers this ratio depends.

We may suppose OA to be changed into OB by the following

processes.

1st. Increase or diminish the length of OA till it becomes equal to that of OB. For this only one number is required, viz. the ratio of the lengths of the two vectors. As Hamilton remarks, this is a positive, or rather a signless, number.

2nd. Turn OA about O until its direction coincides with that of OB, and (remembering the effect of the first

operation) we see that the two vectors now coincide or become identical. To specify this operation three more numbers are required, viz. two angles (such as node and inclination in the case of a planet's orbit) to fix the plane in which the rotation takes place, and one angle for the amount of this rotation.

Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends upon four distinct numbers, whence the name QUATERNION.

48. It is obvious that the operations just described may be performed, with the same result, in the opposite order, being perfectly independent of each other. Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decomposed into two factors of which the order is immaterial.

The stretching factor, or that which performs the first operation in § 47, is called the TENSOR, and is denoted by prefixing T to the quaternion considered.

The turning factor, or that corresponding to the second operation in § 47, is called the VERSOR, and is denoted by the letter U prefixed to the quaternion.

49. Thus, if OA= a, OB = ẞ, and if q be the quaternion which changes a to ẞ, we have

Here it is to be particularly noticed that we write q before a to signify that a is multiplied by q, not q multiplied by a.

[This remark is of extreme importance in quaternions, for,

F

as we shall soon see, the Commutative Law does not generally

apply to the factors of a product.]

We have also, by § 47,

q= TqUq = Uq Tq,

where, as before, Tq depends merely on the relative lengths of a and ẞ, and Uq depends solely on their directions.

Thus, if a, and B, be vectors of unit length parallel to a and ẞ respectively,

Ու

= 1, U

β.

= U

B

a1

50. We must now carefully notice that the quaternion which is the quotient when ẞ is divided by a in no way depends upon the absolute lengths, or directions, of these vectors. Its value will remain unchanged if we substitute for them any other pair of vectors which

(1) have their lengths in the same ratio,

(2) have their common plane the same or parallel, and (3) make the same angle with each other.

B1

(3) LAOB = LA,O,B,.

[Equality of angles is understood to in

clude similarity in direction. Thus the ro

tation about an upward axis is negative (or right-handed) from OA to OB, and also from 0,4, to 0,B,.]

51. The Reciprocal of a quaternion q is defined by the equation,

1

Or, we may reason thus, q changes OA to OB, q1 must therefore change OB to OA, and is therefore expressed by ~(§ 49). B

α

The tensor of the reciprocal of a quaternion is therefore the reciprocal of the tensor; and the versor differs merely by the reversal of its representative angle.

52. The Conjugate of a quaternion q, written Kq, has the same tensor, plane, and angle, only the angle is taken the reverse way. Thus, if

OA OA, and L'OB LAOB,

=

=

Hence qKqKq.q = (Tq)2.

This proposition is obvious, if we recollect

that the tensors of q and Kq are equal, and

that the versors are such that either reverses the effect of the other. The joint effect of these factors is therefore merely to multiply twice over by the common tensor.

53. It is evident from the results of § 50 that, if a and B