CHAP. I.]

RECIPROCAL OF A QUATERNION.

121

defined in 130) remains unchanged, but that the axis (127, 128) is reversed in direction: so that we may write generally,

135. The product of two reciprocal quaternions is always equal to positive unity; and each is equal to the quotient of unity divided by the other; because we have, by 106, 107,

It is therefore unnecessary to introduce any new or peculiar notation, to express the mutual relation existing between a quaternion and its reciprocal; since, if one be denoted by the symbol q, the other may (in the present System, as in Algebra) be denoted by the connected symbol,* 1: q, or -. have thus the two general formulæ (comp. 134):

We

136. Without yet entering on the general theory of multiplication and division of quaternions, beyond what has been done in Art. 120, it may be here remarked that if any two quaternions q and q' be (as in 134) reciprocal to each other, so that q'.q= 1 (by 135), and if q" be any third quaternion, then (as in algebra), we have the general formula,

because if (by 120) we reduce q and q' to a common denominaand denote the new numerators by ẞ and y, we shall have (by the definitions in 106, 107),

tor a,

137. When two complanar triangles AOB, AOB', with a com

The symbol q1, for the reciprocal of a quaternion q, is also permitted in the present Calculus; but we defer the use of it, until its legitimacy shall have been established, in connexion with a general theory of powers of Quaternions.

R

mon side oa, are (as in Fig. 36) inversely similar (118), so that the formula A AOB' ' AOB holds good, then the two unequal

OB

OB'

quotients,* and are said to be CONJUGATE QUATER

ОА

OA

NIONS; and if the first of them be still denoted by q, then the second, which is thus the conjugate of that first, or of any other quaternion which is equal thereto, is denoted by the new symbol, Kq: in which the letter K may be said to be the Characteristic of Conjugation. Thus, with the construction above supposed (comp. again Fig. 36), we may write,

138. From this definition of conjugate quaternions, it follows,

ов'

ов

= K

ОА

OA

hold good, then the line oв'may be

Ist, that if the equation
called (118) the reflexion of the line-OB (and conversely, the latter line
the reflexion of the former), with respect to the line os; IInd, that, under
the same condition, the line oa (prolonged if necessary) bisects per-
pendicularly the line BB', in some point A' (as represented in Fig. 36);
and IIIrd, that any two conjugate quaternions (like any two reciprocal
quaternions, comp. 134, 135) have equal angles, but opposite axes:
so that we may write, generally,

1

139. The reciprocal of a scalar, x, is simply another scalar, or a, having the same algebraic sign, and in all other respects related to x as in algebra. But the conjugate Kx, of a scalar x, considered as a limit of a quaternion, is equal to that scalar x itself; as may be seen by supposing the two equal but opposite angles, AOB and Aов', in Fig. 36, to tend together to

*Compare the Note to page 112.

It will soon be seen that these two last equations (138) express, that the conjugate and the reciprocal, of any proposed quaternion q, have always equal versors, although they have in general unequal tensors.

CHAP. I.] CONjugate and nuLL QUATERNIONS.

123

zero, or to two right angles. We may therefore write, generally,

and conversely*,

Kx=x, if x be any scalar;

=

because then (by 104) we must have OB OB', BB'= 0; and therefore each of the two (now coincident) points, B, B', must be situated somewhere on the indefinite right line oa.

140. In general, by the construction represented in the same Figure, the sum (comp. 6) of the two numerators (or dividend-lines, OB and OB'), of the two conjugate fractions (or quotients, or quaternions), q and Kq (137), is equal to the double of the line oa'; whence (by 106), the sum of those two conjugate quaternions themselves is,

Kq + q = q + Kq = ·

20A'

;

OA

this sum is therefore always scalar, being positive if the angle 4q be acute, but negative if that angle be obtuse.

141. In the intermediate case, when the angle AOB is right, the interval ox' between the origin o and the line BB' vanishes; and the two lately mentioned numerators, OB, OB', become two opposite vectors, of which the sum is null (5). Now, in general, it is natural, and will be found useful, or rather necessary (for consistency with former definitions), to admit that a null vector, divided by an actual vector, gives always a NULL QUATERNION as the quotient; and to denote this null quotient by the usual symbol for Zero. In fact, we have (by 106) the equation,

the zero in the numerator of the left-hand fraction representing here a null line (or a null vector, 1, 2); but the zero on the right-hand side of the equation denoting a null quotient (or quaternion). And thus we are entitled to infer that the sum,

Somewhat later it will be seen that the equation Kq= q may also be written as Vq=0; and that this last is another mode of expressing that the quaternion, q, degenerates (131) into a scalar.

Kq+q, or q + Kq, of a right-angled quaternion, or right quotient (132), and of its conjugate, is always equal to zero.

142. We have, therefore, the three following formulæ, whereof the second exhibits a continuity in the transition from the first to the third:

And because a quaternion, or geometric quotient, with an actual and finite divisor-line (as here oA), cannot become equal to zero unless its dividend-line vanishes, because (by 104) the equation

B

α

if a be any actual and finite vector, we may infer, conversely, that the sum qKq cannot vanish, without the line OA' also vanishing; that is, without the lines OB, ов' becoming opposite vectors, and therefore the quaternion q becoming a right quotient (132). We are therefore entitled to establish the three following converse formulæ (which indeed result from the three former):

143. When two opposite vectors (1), as ß and – ß, are both divided by one common (and actual) vector, a, we shall say that the two quotients, thus obtained are OPPOSITE QUATERNIONS; so that the opposite of any quaternion q, or of any quotient B: a, may be denoted as follows (comp. 4):

CHAP. I.]

OPPOSITE QUATERNIONS.

125

while the quaternion q itself may, on the same plan, be denoted (comp. 7) by the symbol 0+q, or + q. The sum of any two opposite quaternions is zero, and their quotient is negative unity; so that we may write, as in algebra (comp. again 7), (− q) + q = (+ q) + (−q)=0; (−q): q=-1; −q=(- 1)q ; because, by 106 and 141,

The reciprocals of opposite quaternions are themselves opposite; or in symbols (comp. 126),

Opposite quaternions have opposite axes, and supplementary angles (comp. Fig. 33, bis); so that we may establish (comp. 132, (5.) ) the two following general formulæ,

L (− q) = π- 293 Ax.(q)=- Ax.q.

144. We may also now write, in full consistency with the recent formulæ II. and II'. of 142, the equation,

In words, the conjugate of a right quotient, or of a right-angled (or right) quaternion (132), is the right quotient opposite thereto; and conversely, if an actual quaternion (that is, one which is not null) be opposite to its own conjugate, it must be a right quotient.

we shall know that pa; and therefore (if a = OA, and p=OP, as before), that the

*It will be seen at a later stage, that the equation Kq-q, or q + Kq = 0, may be transformed to this other equation, Sq=0; and that, under this last form, it expresses that the scalar part of the quaternion q vanishes: or that this quaternion is a right quotient (132).