(i. e. the versor which turns a westward unit-vector into an upward one will turn the upward into an eastward unit);

or K=j(-1) = −jI*. .

(10)

Now let us operate on the two equal vectors in (10) by the same versor, i, and we have

The meaning of these important equations is very simple; and is, in fact, obvious from our construction in § 54 for the multiplication of versors; as we see by the annexed figure, where we must remember that i, j, k are quadrantal versors whose planes are at right angles, so that the figure represents a hemisphere divided into quadrantal triangles.

W

N

70.] But, by the same figure,

iON = OZ,

whence ji ON=jOZ OE-OW‒‒k ON.

=

71.] From this it appears that

=

* The negative sign, being a mere numerical factor, is evidently commutative with j; indeed we may, if necessary, easily assure ourselves of the fact that to turn the negative (or reverse) of a vector through a right (or indeed any) angle, is the same thing as to turn the vector through that angle and then reverse it.

These equations, along with

i2 = j2 = k2 = -1 ((7), (8), (9)),

contain essentially the whole of Quaternions. But it is easy that, for the first group, we may substitute the single equation

to see

(13)

since from it, by the help of the values of the squares of i, j, k, all the other expressions may be deduced. We may consider it proved in this way, or deduce it afresh from the figure above, thus

72.] One most important step remains to be made, to wit the assumption referred to in § 64. We have treated i, j, k simply as quadrantal versors; and I, J, K as unit-vectors at right angles to each other, and coinciding with the axes of rotation of these versors. But if we collate and compare the equations just proved we have

with the other similar groups symmetrically derived from them. Now the meanings we have assigned to i, j, k are quite independent of, and not inconsistent with, those assigned to I, J, K. And it is superfluous to use two sets of characters when one will suffice. Hence it appears that i, j, k may be substituted for I, J, K; in other words, a unit-vector when employed as a factor may be considered as a quadrantal versor whose plane is perpendicular to the vector. This is one of the main elements of the singular simplicity of the quaternion calculus.

73.] Thus the product, and therefore the quotient, of two perpendicular vectors is a third vector perpendicular to both.

Hence the reciprocal (§ 51) of a vector is a vector which has the opposite direction to that of the vector, and its length is the reciprocal of the length of the vector.

The conjugate (§ 52) of a vector is simply the vector reversed. Hence, by § 52, if a be a vector

74.] We may now see that every versor may be represented by a power of a unit-vector.

For, if a be any vector perpendicular to i (which is any definite unit-vector),

ia, B, is a vector equal in length to a, but perpendicular to both i and a;

i2a = - -a,

i3 a―ia =—ß,

¿1a iß-i2 a = a.
= - - :

Thus, by successive applications of i, a is turned round i as an axis through successive right angles. Hence it is natural to define im as a versor which turns any vector perpendicular to i through m right angles in the positive direction of rotation about i as an axis. Here m may have any real value whatever, whole or fractional, for it is easily seen that analogy leads us to interpret a negative value of m as corresponding to rotation in the negative direction.

75.] From this again it follows that any quaternion may be expressed as a power of a vector. For the tensor and versor elements of the vector may be so chosen that, when raised to the same power, the one may be the tensor and the other the versor of the given quaternion. The vector must be, of course, perpendicular to the plane of the quaternion.

76.] And we now see, as an immediate result of the last two sections, that the index-law holds with regard to powers of a quaternion (§ 63).

77.] So far as we have yet considered it, a quaternion has been regarded as the product of a tensor and a versor : we are now to consider it as a sum. The easiest method of so analysing it seems

OB

Let represent any quaternion. Draw
OA

BC perpendicular to 04, produced if neces

sary.

Then, § 19,

But, § 22,

OB = OC+CB.

OC = x0A,

where is a number, whose sign is the same as that of the cosine of ▲ AOB.

Also, § 73, since CB is perpendicular to OA,

CB = yбA,

where γ is a vector perpendicular to OA and CB, i.e. to the plane of the quaternion.

Hence

OB

OA

= x+y.

Thus a quaternion, in general, may be decomposed into the sum of two parts, one numerical, the other a vector. Hamilton calls them the SCALAR, and the VECTOR, and denotes them respectively by the letters S and V prefixed to the expression for the quaternion. 78.] Hence q = Sq+Vq, and if in the above example

79.] If, in the figure of last section, we produce BC to D, so as to double its length, and join OD, we have, by § 52,

Comparing this value of OC with that in last section, we find

or the scalar of the conjugate of a quaternion is equal to the scalar of the quaternion.

Again, CD = -CB by the figure, and the substitution of their values gives

V Kq = -Vq,

(2)

or the vector of the conjugate of a quaternion is the vector of the quaternion reversed.

We may remark that the results of this section are simple consequences of the fact that the symbols S, V, K are commutative †. Thus

SKq=KSq= Sq,

since the conjugate of a number is the number itself; and

VKq = KVq = −Vq (§ 73).

* The points are inserted to shew that S and V apply only to q, and not to q04. It is curious to compare the properties of these quaternion symbols with those of the Elective Symbols of Logic, as given in BOOLE's wonderful treatise on the Laws of Thought; and to think that the same grand science of mathematical analysis, by processes remarkably similar to each other, reveals to us truths in the science of position far beyond the powers of the geometer, and truths of deductive reasoning to which unaided thought could never have led the logician.

Again, it is obvious that

and thence

Στη = S2q, ΣVq = V£q,

ΣKq: = ΚΣΥ.

80.] Since any vector whatever may be represented by

xi+yj+zk

where x, y, z are numbers (or Scalars), and i, j, k may be any three non-coplanar vectors, §§ 23, 25-though they are usually understood as representing a rectangular system of unit-vectors—and since any scalar may be denoted by w; we may write, for any quaternion q, the expression

q=w+xi+yj+zk (§ 78).

Here we have the essential dependence on four distinct numbers, from which the quaternion derives its name, exhibited in the most simple form.

And now we see at once that an equation such as

where

q = q,
q=w′+x'i+yj+Zk,

involves, of course, the four equations

w= w; x = x, y = y, z=z.

81.] We proceed to indicate another mode of proof of the distributive law of multiplication.

We have already defined, or assumed (§ 61), that

and have thus been able to understand what is meant by adding two quaternions.

But, writing a for a-1, we see that this involves the equality (B+y) a = Ba+ya;

from which, by taking the conjugates of both sides, we derive a' (B′+ y) = a' ́ß' + áŸ'′ (§ 55).

And a combination of these results (putting ẞ+y for a' in the latter, for instance) gives

(B+y) (B′+ y) = (B+ y) B′ + (B + y) y

= BB'+YB'+By+yy by the former.

Hence the distributive principle is true in the multiplication of vectors. It only remains to shew that it is true as to the scalar and