124. With the same notation for complanarity, we may write generally,

xa |||a, ß;

a and ẞ being any two vectors, and x being any scalar; because, if a = OA and B = Oв as before, then (by 15, 17) xα = oa', where a' is some point on the indefinite right line through the points o and A: so that the plane AOB contains the line oa'. For a similar reason, we have generally the following formula of complanarity of quotients,

YB B

Ха a

whatever two scalars x and y may be; a and ẞ still denoting any two vectors.

125. It is evident (comp. Fig. 35) that if ▲ AOB & COD, then A BOA & DOC,

and A AOC & BOD;

whence it is easy to infer that for quaternions, as well as for ordinary or algebraic quotients,

it being permitted now to establish the converse of the last formula of 118, or to say that

Under the same condition, by combining inversion with alter

nation, we have also this other equation,

α β

=

γ

B

126. If the sides, OA, OB, of a triangle AOB, or those sides either way prolonged, be cut (as in Fig. 37) by any parallel, A'B' or a′′B", to the base AB, we have evidently the relations of direct similarity (118),

▲ A'OB' ∞ AOB, ▲ A"OB" ∞ AOB; whence (comp. Art. 13 and Fig. 12) it follows that we may write, for quaternions as in algebra, the general B" equation, or identity,

B'

A"

Ο

Α'

A

Fig. 37.

CHAP. I.] AXIS AND ANGLE OF A QUATERNION.

αβ β

= ; xa a

117

where x is again any scalar, and a, ẞ are any two vectors. It is easy also to see, that for any quaternion q, and any scalar x, we have the product (comp. 107),

so that, in the multiplication of a quaternion by a scalar (as in the multiplication of a vector by a scalar, 15), the order of the factors is indifferent.

SECTION 5. On the Axis and Angle of a Quaternion; and on

the Index of a Right Quotient, or Quaternion.

127. From what has been already said (111, 112), we are naturally led to define that the Axis, or more fully that the positive axis, of any quaternion (or geometric quotient) OB: OA, is a right line perpendicular to the plane AOB of that quaternion; and is such that the rotation round this axis, from the divisorline os, to the dividend-line OB, is positive: or (as we shall henceforth assume) directed towards the right-hand,* like the motion of the hands of a watch.

128. To render still more definite this conception of the axis of a quaternion, we may add, Ist, that the rotation, here spoken of, is supposed (112) to be the simplest possible, and therefore to be in the plane of the two lines (or of the quaternion), being also generally less than a semi-revolution in that plane; IInd, that the axis shall be usually supposed to be a line ox drawn from the assumed origin o; and IIIrd, that the length of this line shall be supposed to be given, or fixed, and to be equal to some assumed unit of length: so that the term x, of this axis ox, is situated (by its construction) on a given spheric surface described about the origin o as centre, which surface we may call the surface of the UNIT-SPHERE.

129. In this manner, for every given non-scalar quotient

This is, of course, merely conventional, and the reader may (if he pleases) substitute the left-hand throughout.

(108), or for every given quaternion q which does not reduce itself (or degenerate) to a mere positive or negative number, the axis will be an entirely definite vector, which may be called an UNIT-VECTOR, on account of its assumed length, and which we shall denote*, for the present, by the symbol Ax.q. Employing then the usual sign of perpendicularity, 1, we may now write, for any two vectors a, Bß, the formula:

130. The ANGLE of a quaternion, such as OB: OA, shall simply be, with us, the angle AOB between the two lines, of which the quaternion is the quotient; this angle being supposed here to be one of the usual kind (such as are considered by Euclid) and therefore being acute, or right, or obtuse (but not of any class distinct from these), when the quaternion is a non-scalar (108). We shall denote this angle of a quaternion q, by the symbol, q; and thus shall have, generally, the two inequalities following:

where is used as a symbol for two right angles.

131. When the general quaternion, q, degenerates into a scalar, x, then the axis (like the planet) becomes entirely indeterminate in its direction; and the angle takes, at the same time, either zero or two right angles for its value, according as the scalar is positive or negative. Denoting then, as above, any such scalar by x, we have:

At a later stage, reasons will be assigned for denoting this axis, Ax.q, of a quaternion q, by the less arbitrary (or more systematic) symbol, UVq; but for the present, the notation in the text may suffice.

In some investigations respecting complunar quaternions, and powers or roots of quaternions, it is convenient to consider negative angles, and angles greater than two right angles: but these may then be called AMPLITUDES; and the word Angle," like the word "Ratio," may thus be restricted, at least for the present, to its ordinary geometrical sense.

Compare the Note to page 114. The angle, as well as the axis, becomes indeterminate, when the quaternion reduces itself to zero; unless we happen to know a law, according to which the dividend-line tends to become null, in the transition B 0

from to

CHAP. I.] CASE OF A RIGHT QUOTIENT, OR QUATERNION.

Ax. r an indeterminate unit-vector;

≤ x = 0, if x > 0; x = π, if x < 0.

132. Of non-scalar quaternions, the most im- B portant are those of which the angle is right, as in the annexed Figure 38; and when we have thus,

the quaternion q may then be said to be a RIGHT QUOTIENT; or sometimes, a Right Quaternion.

119

Fig. 38.

(1.) If then a = OA and p=OP, where o and a are two given (or fixed) points, but P is a variable point, the equation

π

<==
a 2

expresses that the locus of this point P is the plane through 0, perpendicular to the line OA; for it is equivalent to the formula of perpendicularity pa (129). (2.) More generally, if ẞ= OB, B being any third given point, the equation,

expresses that the locus of P is one sheet of a cone of revolution, with o for verter, and OA for axis, and passing through the point в; because it implies that the angles AOB and AOP are equal in amount, but not necessarily in one common plane.

(3.) The equation (comp. 128, 129),

B
Ax. = Ax.

a

expresses that the locus of the variable point P is the given plane AOB; or rather the indefinite half-plane, which contains all the points P that are at once complanar with the three given points o, A, B, and are also at the same side of the indefinite right line os, as the point B.

expresses that the point P is situated, either on the finite right line ox, or on that line prolonged through, but not through 0; so that the locus of P may in this case be said to be the indefinite half-line, or ray, which sets out from o in the direction of the rector OB or ẞ; and we may write p=x3, x>0 (x being understood to be a scalar), instead of the equations assigned above.

* Reasons will afterwards be assigned, for equating such a quotient, or quaternion, to a Vector; namely to the line which will presently (133) be called the Index of the Right Quotient.

(6.) Other notations, for representing these and other geometric loci, will be found to be supplied, in great abundance, by the Calculus of Quaternions; but it seemed proper to point out these, at the present stage, as serving already to show that even the two symbols of the present Section, Ax. and 4, when considered as Characteristics of Operation on quotients of vectors, enable us to express, very simply and concisely, several useful geometrical conceptions.

133. If a third line, o1, be drawn in the direction of the axis ox of such a right quotient (and therefore perpendicular, by 127, 129, to each of the two given rectangular lines, oa, OB); and if the length of this new line or bear to the length of that axis ox (and therefore also, by 128, to the assumed unit of length) the same ratio, which the length of the dividendline, OB, bears to the length of the divisor-line, oa; then the line or, thus determined, is said to be the INDEX of the Right Quotient. And it is evident, from this definition of such an Index, combined with our general definition (117, 118) of Equality between Quaternions, that two right quotients are equal or unequal to each other, according as their two indexlines (or indices) are equal or unequal vectors.

SECTION 6.- On the Reciprocal, Conjugate, Opposite, and Norm of a Quaternion; and on Null Quaternions.

134. The RECIPROCAL (or the Inverse, comp. 119) of a

quaternion, such as q = B, is that other quaternion,

a

a

which is formed by interchanging the divisor-line and the dividend-line; and in thus passing from any non-scalar quaternion to its reciprocal, it is evident that the angle (as lately