CHAP. I.]

TENSOR OF A QUATERNION.

169

from OA' to OB, as Fig. 48 may serve to illustrate; or, IInd, to begin by stretching, from oa to OB', and end by turning, from oв' to OB. The act of multiplication of a line a by a quaternion q, considered as a factor (103), which affects both length and direction (109), may thus be decomposed into two distinct and partial acts, of the kinds which we have called Version and Tension; and these two acts may be performed, at pleasure, in either of two orders of succession. And although, if we attended merely to lengths, we might be led to say that the tensor of a quaternion was a signless number,* expressive of a geometrical ratio of magnitudes, yet when the recent construction (Fig. 48) is adopted, we see, by either of the two resulting expressions (188) for Tq, that there is a propriety in treating this tensor as a positive scalar, as we have lately done, and propose systematically to do.

190. Since TKq = Tq, by 187, (12.), and UKq=1: Uq, by 158, we may write, generally, for any quaternion and its conjugate, the two connected expressions:

whence, by multiplication and division,

III... q. Kq= (Tq)2; IV... q: Kq=(Uq).

This last formula had occurred before; and we saw (161) that in it the parentheses might be omitted, because (Uq)=U(q). In like manner (comp. 161, (2.) ), we have also

parentheses being again omitted; or in words, the tensor of the square of a quaternion is always equal to the square of the tensor: as appears (among other ways) from inspection of Fig. 42, bis, in which the lengths of OA, OB, oc form a geometrical progression; whence

At the same time, we see again that the product qKq of two conju gate quaternions, which has been called (145, (11.)) their common Norm, and denoted by the symbol Nq, represents geometrically the square of the quotient of the lengths of the two lines, of which (when considered as vectors) the quaternion q is itself the quotient (112). therefore write generally,†

We

may

V... qKq=Tq2=Nq;

VI... Tq Nq= √(qKq).

* Compare the Note in page 108, to Art. 109.
Compare the Note in page 129.

Z

(1.) We have also, by II., the following other general transformations for the tensor of a quaternion :

of which the geometrical significations might easily be exhibited by a diagram, but of which the validity is sufficiently proved by what precedes.

(3.) The reciprocal of a quaternion, and the conjugate of that reciprocal, may now be thus expressed:

191. In general, let any two quaternions, q and q', be considered as multiplicand and multiplier, and let them be reduced (by 120) to the forms ẞ: a and y: ß; then the tensor and versor of that third quaternion, y: a, which is (by 107) their product qq, may be thus expressed :

=

I...Tq'q=T(y: a) = Ty: Ta = (Ty: Tß).(Tß: Ta) = Tq'. Tq ; II... Ugg U(y: a)=Uy: Ua=(Uy: Uß). (Uß: Ua)=Uq.Uq; where Tag and Uq'q are written, for simplicity, instead of T(q.q) and U(q'.q). Hence, in any such multiplication, the tensor of the product is the product of the tensor; and the versor of the product is the product of the versors; the order of the factors being generally retained for the latter (comp. 168, &c.), although it may be varied for the former, on account of the scalar character of a tensor. In like manner, for the division of any one quaternion q', by any other q, we have the analogous formulæ :

III... T(qq) = Tg': Tq; IV... U(q' q) = Ug': Uq; or in words, the tensor of the quotient of any two quaternions is equal to the quotient of the tensors; and similarly, the versor of the quotient is equal to the quotient of the versors. And because multiplication and division of tensors are performed according to the rules of algebra, or rather of arithme

Compare Art. 145, and the Note to page 127.

CHAP. I.] PRODUCT OR QUOTIENT OF TWO QUATERNIONS.

171

tie (a tensor being always, by what precedes, a positive number), we see that the difficulty (whatever it may be) of the general multiplication and division of quaternions is thus reduced to that of the corresponding operations on versors: for which latter operations geometrical constructions have been assigned, in the ninth Section of the present Chapter.

(1.) The two products, q'q and qq', of any two quaternions taken as factors in two different orders, are equal or unequal, according as those two factors are complanar or diplanar; because such equality (169), or inequality (168), has been already proved to exist, for the case when each tensor is unity: but we have always (comp. 178),

(2.) If Lq = Lq' :

π

Tq'q = Tgq', and Lg'q=Lgg'.

2'

then qq' =Kq'q (170); so that the products of two right

quotients, or right quaternions (132), taken in opposite orders, are always conjugate quaternions.

so that the product of two right quaternions, in two rectangular planes, is a third right quaternion, in a plane rectangular to both; and is changed to its own opposite, when the order of the factors is reversed: as we had ij=k=-ji (182).

(4.) In general, if q and q' be any two diplanar quaternions, the rotation round Ax. q, from Ax. q to Ax. q'q, is positive (177).

(5.) Under the same condition, q(q′ : q) is a quaternion with the same tensor, and same angle, as q', but with a different axis; and this new axis, Ax.q(q': q), may be derived (179, (1.)) from the old axis, Ax.q', by a conical rotation (in the positive direction) round Ax. q, through an angle = 2 ▲ q.

(6.) The product or quotient of two complanar quaternions is, in general, a third quaternion complanar with both; but if they be both scalar, or both right, then this product or quotient degenerates (131) into a scalar.

(7.) Whether q and q' be complanar or diplanar, we have always as in algebra (comp. 106, 107, 136) the two identical equations:

V... (q' : q). q = q' ;

VI... (g' · g) : q = q'.

(8.) Also, by 190, V., and 191, I., we have this other general formula:

or in words, the norm of the product is equal to the product of the norms.

192. Let q=ẞ: a, and q'=y: ẞ, as before; then

1 : q'q = 1 : (y : a) = a : y = (a : ẞ). (B : y) = (1 : q) . (1: q'); so that the reciprocal of the product of any two quaternions is

equal to the product of the reciprocals, taken in an inverted order: or briefly,

I... Rq'q = Rq. Rq',

1

if R be again used (as in 161, (3.)) as a (temporary) characteristic of reciprocation. And because we have then (by the same sub-article) the symbolical equation, KUUR, or in words, the conjugate of the versor of any quaternion q is equal (158) to the versor of the reciprocal of that quaternion; while the versor of a product is equal (191) to the product of the versors: we see that

But

KUq'q = URq'q = URq. URq' = KUq. KUq'.

Kq= Tq. KUq, by 190, IX.; and Tq'q = Tg'. Tq = Tq.Tq', by 191; we arrive then thus at the following other important and general formula:

II. . . Kq'q = Kq. Kq' ;

or in words, the conjugate of the product of any two quaternions is equal to the product of the conjugates, taken (still) in an inverted order.

(1.) These two results, I., II., may be illustrated, for versors (Tq=Tq' = 1), by the consideration of a spherical triangle ABC (comp. Fig. 43); in which the sides AB and BC (comp. 167) may represent q and q', the arc AC then representing q'q. For then the new multiplier Rq=Kq (158) is represented (162) by BA, and the new multiplicand Rq' = Kq' by CB; whence the new product, Rq. Rq' = Kq. Kq', is represented by the inverse arc CA, and is therefore at once the reciprocal Rq'q, and the conjugate Kq'q, of the old product qʻq.

(2.) If q and q' be right quaternions, then Kq=-q, Kg':

=

q' (by 144); and

the recent formula II. becomes, Kq'q=qq', as in 170.
(3.) In general, that formula II. (of 192) may be thus written :

CHAP. I.] CASE OF TWO RIGHT QUATERNIONS.

173

so that this third formula of inverse similitude is a consequence from the other two. (5.) If then (comp. 145, (6.)) any two circles, whether in one plane or in space, touch one another at a point в; and if from any point o, on the common tangent BO, two secants OAC, OED be drawn, to these two circles; the four points of section, A, C, D, E, will be on one common circle: for such concircularity is an easy consequence (through equal angles, &c.), from the last inverse similitude.

(6.) The same conclusion (respecting concircularity, &c.) may be otherwise and geometrically drawn, from the equality of the two rectangles, AOC and DOE, each being equal to the square of the tangent OB; which may serve as an instructive verification of the recent formula III., and as an example of the consistency of the results, to which calculations with quaternions conduct.

(7.) It may be noticed that the construction would in general give three circles, although only one is drawn in the Figure; but that if the two triangles ABC and DBE be situated in different planes, then these three circles, and of course the five points ABCDE, are situated on one common sphere.

193. An important application of the foregoing general theory of Multiplication and Division, is to the case of Right Quaternions (132), taken in connexion with their Index- Vectors, or Indices (133).

Considering division first, and employing the general formula of 106, let ẞ and y be each a; and let ẞ'and y' be the respective indices of the two right quotients, q=ß: a, and q=y:a. We shall thus have the two complanarities, B'll ẞ, y, and y' ẞ, y (comp. 123), because the four lines ẞ, y, B', y' are all perpendicular to a; and within their common plane it is easy to see, from definitions already given, that these four lines form a proportion of vectors, in the same sense in which u, B, y, & did so, in the fourth Section of the present Chapter: so that we may write the equation of quotients,

In fact, we have (by 133, 185, 187) the following relations of length,

=

TB TB: Ta, Ty' = Ty: Ta, and .. T(y': B') = T(y: ẞ); while the relation of directions, expressed by the formula,

U(y': B') = U(y: ẞ), or Uy: Uß' = Uy : Uß,

is easily established by means of the equations,