List of Works referred to in the following pages.

1. Lectures on Quaternions. By Sir W. R. Hamilton. Dublin, 1853. 2. Elements of Quaternions. By Sir W. R. Hamilton. Dublin, 1866. 3. Introduction to Quaternions. By Professors P. Kelland and P. G. Tait. London, 1873.

4. Kurze Anleitung zum Rechnen mit den Hamilton'schen Quaternionen. By Professor J. Odstrčil. Halle a. S., 1879.

5. Elements of Quaternions. By Professor A. S. Hardy. Boston, U.S., 1881.

6. Mathematical Papers. By Professor W. K. Clifford.

7. Elementary Treatise on Quaternions. 3rd. ed., Cambridge, 1890.

London, 1882.

By Professor P. G. Tait.

8. Theorie der Quaternionen. By Dr. P. Molenbroek. Leiden, 1891.

THE

OUTLINES OF QUATERNIONS

PART I

SUBTRACTION AND ADDITION OF VECTORS

CHAPTER I

FIRST PRINCIPLES OF VECTORS

SECTION 1

The Nature of a Vector

1. Definition. A Vector is any quantity which has Magnitude and Direction (Clifford).

It follows that a straight line, AB, considered as having not only length but direction, is a vector. Its initial point, A, is called its Origin; and its final point, B, is called its Term.

With the exception of three special vectors (i, j, k, Pt. II., 6°), vectors will be denoted in these pages either by a symbol combining their initial and final letters, such as AB, or by a small letter of the Greek alphabet, in order to distinguish them from the ordinary straight lines of geometry, such as AB or a.

2o. A vector, AB, may be conceived as having for its function to transport (vehere, to carry) a particle from A to B. A vector thus implies an operation, and represents translation in a certain direction for a certain distance.

B

3°. When its origin and term, A and B, are distinct points, AB is said to be an Actual Vector; but when, as a limit, these points coincide, it is said to be a Null Vector.

Actual is used as opposed to null; real as opposed to imaginary.

4°. In order to determine the position of any point in space, B, in relation to any other point, A, three numbers must be known. Let A be the centre of the earth (supposed to be a perfect sphere), and B any point upon its surface. Then, in order to be able to draw a straight line from A to B we must know, first, the Latitude of B; secondly, its Longitude; and thirdly, the Radius of the Earth.

Every vector, then, implicitly involves three numbers; one indicating its length, and two its direction.

5°. A vector is not to be confounded with the radius vector of Algebraic Geometry. The latter represents length only, and implies but one number. It is, in fact, one of the three numbers contained in a vector.

6°. Opposite Vectors, such as AB and BA, are sometimes called Vector and Revector.

Coinitial Vectors are vectors whose origins coincide.

If there be any series of vectors such that the origin of the second coincides with the term of the first, the origin of the third with the term of the second, &c., &c., these vectors are called Successive Vectors.

Coplanar Vectors are those that lie in the same plane. Diplanar Vectors are those that lie in different planes.

We will have hereafter to consider vector arcs; but at present the only vectors considered are rectilinear.

SECTION 2

Equality and Inequality of Vectors

7°. Definition.-Two given vectors are equal to each other when (and only when) the origin and term of the one can be brought to coincide simultaneously with the corresponding points of the other, by motion of translation, without rotation.

As a consequence of this definition, no two vectors are equal unless they have, first, equal lengths, and, secondly, similar directions the phrase 'similar directions' meaning 'parallel directions with the same sense.' Similarly, 'contrary

(or opposite) directions' means 'parallel directions with con trary (or opposite) sense.'

The meaning of the word 'parallel' is extended, so as to include lines which form parts of one common straight line.

8°. If two equal vectors, AB and CD, do not form part of one common straight line, they may be regarded as the opposite sides of a parallelogram, ACDB, fig. 1.

C

FIG. 1.

B

9°. Since the operation implied by a vector-transference in a certain direction for a certain distance-is the same, whatever point in space be selected as the origin of motion; all equal vectors are denoted by the same vector-symbol. Thus, if AB = CD, and if AB be denoted by ß, CD is also denoted by ß. It follows that a (Hamiltonian) vector has no particular position in space.

SECTION 3

Subtraction and Addition of Two Vectors

10°. Definition. When a first vector, AB, is subtracted from a second vector, AC, which is coinitial with it, or from a third vector, A'C', which is equal to that second vector, the remainder is that fourth vector, BC, which is drawn from the term B of the first to the term С of the second vector (Hamilton).

In symbols, fig. 1,

The foregoing definition is perfectly general, and includes the case in which the vectors are parallel, i.e. in which Z CAB

=π, or zero.

If AC be a null vector, the equation AC AB BC becomes

Therefore, the minus sign reverses the direction of a vector; and if AB is represented by a, BA will be represented by

-a.

where AC is said to be added to BA.

Or, if BA and AC be successive vectors, 6o, their sum is a vector, BC, drawn from the origin of the first, B, to the term of the second, C.

Hence,

BA + AB = BB = 0

(1)

12°. We have now to consider the sum of two nonsuccessive vectors.

Definition. If there be two successive vectors, AC and CD, and if a third vector, C'D', be equal to the second, but not successive to the first; the sum obtained by adding the third to the first is that fourth vector, AC, which is drawn from the origin of the first to the term of the second (Hamilton).

In symbols, fig. 1,

C'D' AC CD + AC = AD.

This definition holds good when the vectors are parallel, i.e, when ACD = π or zero.

If CD be a null vector,

+ AC AC; or, + AB = AB.

13°. By 10° and 12° we have:

it follows that a vector may be tranferred from one side of an equation to the other by changing its sign.

it follows that directions can be assigned to the sides of any