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And thus the mean proportional between two vectors (in the given plane) becomes, in all cases, determined: at least if their order (as first and third) be given.
(3.) If the restriction (225) on the common plane of the lines, were removed, we might then, on the recent plan (227), fix definitely the direction, as well as the length, of the mean OB, in every case but one: this excepted case being that in which, as in (2.), the two given extremes, OA, OC, have exactly opposite directions; so that the angle (AOC = π) between them has no one definite bisector. In this case, the sought point B would have no one determined position, but only a locus: namely the circumference of a circle, with o for centre, and with a radius equal to the geometric mean between OA, OC, while its plane would be perpendicular to the given right line AOC. (Comp. again the Figures 41; and the remarks in 148, 149, 153, 154, on the square of a right radial, or versor, and on the partially indeterminate character of the square root of a negative scalar, when interpreted, on the plan of this Calculus, as a real in geometry.)
228. The quotient of any two complanar and right quaternions has been seen (191, (6.)) to be a scalar; since then we here suppose (225) that q |||i, we are at liberty to write,
I. . . Sq = x;
Vq: i = y;
Vq = yi = iy;
and consequently may establish the following Reduction of a Quaternion in the given Plane (of i) to a Standard Binomial Form (comp. 221):
II. . . q = x + iy, if q|||i;
x and y being some two scalars, which may be called the two constituents (comp. again 221) of this binomial. And then an equation between two quaternions, considered as binomials of this form, such as the equation,
breaks up (by 202, (5.)) into two scalar equations between their respective constituents, namely,
IV... x' = x, y = y,
notwithstanding the geometrical reality of the right versor, i.
(1.) On comparing the recent equations II., III., IV., with those marked as III., V., VI., in 221, we see that, in thus passing from general to complanar quaternions, we have merely suppressed the coefficients of j and k, as being for our present purpose, null; and have then written x and y, instead of w and x.
It is permitted, by 227, XI., to write this expression as x+y V-1; but the form xiy is shorter, and perhaps less liable to any ambiguity of interpretation.
STANDARD BINOMIAL FORM, COUPLE.
(2.) As the word "binomial" has other meanings in algebra, it may be convenient to call the form II. & COUPLE; and the two constituent scalars x and y, of which the values serve to distinguish one such couple from another, may not unnaturally be said to be the Co-ordinates of that Couple, for a reason which it may be useful to state.
(3.) Conceive, then, that the plane of Fig. 50 coincides with that of i, and that positive rotation round Ax.i is, in that Figure, directed towards the left-hand; which may be reconciled with our general convention (127), by imagining that this axis of i is directed from o towards the back of the Figure; or below* it, if horizontal. This being assumed, and perpendiculars BB', BB" being let fall (as in the Figure) on the indefinite line oa itself, and on a normal to that line at o, which normal we may call oa', and may suppose it to have a length equal to that of oa, with a left-handed rotation AOA', so that
V... oa'= i. oa, or briefly, V'... a' = ia,
while BOB', and ẞ"= OB", as in 201, and q = ß: a, as in 202;
then, on whichever side of the indefinite right line OA the point в may be situated, a comparison of the quaternion q with the binomial form II. will give the two equations,
VI... x (= Sq) = ẞ': a;
y (=Vq: i=ẞ": ia) = ß" : a' ;
so that these two scalars, x and y, are precisely the two rectangular co-ordinates of the point B, referred to the two lines oa and oa', as two rectangular unit-axes, of the ordinary (or Cartesian) kind. And since every other quaternion, q' = x'+iy', in the given plane, can be reduced to the form y: a, or OC: OA, where c is a point in that plane, which can be projected into c' and c" in the same way (comp. 197, 205), we see that the two new scalars, or constituents, r' and y', are simply (for the same reason) the co-ordinates of the new point c, referred to the same pair of
(4.) It is evident (from the principles of the foregoing Chapter), that if we thus express as couples (2.) any two complanar quaternions, q and q', we shall have the following general transformations for their sum, difference, and product :
VII. q'±q= (x' + x)+i (y'±y);
VIII... q'. q = (x'x − y'y) + i (x'y+y'x).
(5.) Again, for any one such couple, 7, we have (comp. 222) not only Sq = x, and Vqiy, as above, but also,
(6.) Hence, for the quotient of any two such couples, we have,
(7.) The law of the norms (191, (8.)), or the formula, Ng'q = Ng'. Ng, is expressed here (comp. 222, (3.)) by the well-known algebraic equation, or identity, XV. . . (x22 + y′2) (x2 + y2) = (x'x − y'y)2 + (x'y + y'x)2 ;
in which xyx'y' may be any four scalars.
SECTION 2.-On Continued Proportion of Four or more Vectors; Whole Powers and Roots of Quaternions; and Roots of Unity.
229. The conception of continued proportion (227) may easily be extended from the case of three to that of four or more (complanar) vectors; and thus a theory may be formed of cubes and higher whole powers of quaternions, with a correspondingly extended theory of roots of quaternions, including roots of scalars, and in particular of unity. Thus if we suppose that the four vectors aßyd form a continued proportion, expressed by the formula,
(by an obvious extension of usual algebraic notation,) we may say that the quaternion 8: a is the cube, or the third power, of B:a; and that the latter quaternion is, conversely, a cuberoot (or third root) of the former; which last relation may naturally be denoted by writing,
230. But it is important to observe that as the equation qQ, in which q is a sought and Q is a given quaternion, was found to be satisfied by two opposite quaternions q, of the form Q (comp. 227, VII.), so the slightly less simple equation q3 = Q is satisfied by three distinct and real quaternions, if Q be actual and real; whereof each, divided by either of the other two, gives for quotient a real quaternion, which is equal to one of the cube-roots of positive unity. In fact, if we conceive (comp. the annexed Fig. 54) that ẞ' and ẞ" are two other but equally long vectors in the given plane, ob
CHAP. II.] CUBE-ROOTS OF A QUATERNION, AND OF UNITY. 247
tained from ẞ by two successive and positive rotations, each through the third part of a circumference,
so that we are equally entitled, at this stage, to write, instead of III. or III'., these other equations:
231. A (real and actual) quaternion Q may thus be said to have three (real, actual, and) distinct cube-roots; of which however only one can have an angle less than sixty degrees; while none can have an angle equal to sixty degrees, unless the proposed quaternion Q degenerates into a negative scalar. In every other case, one of the three cube-roots of Q, or one of the three values of the symbol Q, may be considered as simpler than either of the other two, because it has a smaller angle (comp. 199, (1.)); and if we, for the present, denote this one, which we shall call the Principal Cube-Root of the quaternion Q, by the symbol Q, we shall thus be enabled to establish the formula of inequality,
VIII... <Q<, if Q<T.
232. At the limit, when Q degenerates, as above, into a negative scalar, one of its cube-roots is itself a negative scalar, and has there
fore its angle; while each of the two other roots has its angle
In this case, among these two roots of which the angles are equal to each other, and are less than that of the third, we shall consider as simpler, and therefore as principal, the one which answers (comp. 227, (2.)) to a positive rotation through sixty degrees; and so shall be led to write,
using thus the positive sign for the radical 3, by which i is multiplied in the expression IX. for 2/-1; with the connected formula,
although it might at first have seemed more natural to adopt as principal the scalar value, and to write thus,
which latter is in fact one value of the symbol, (− 1)*.
(1.) We have, however, on the present plan, as in arithmetic,
can be verified in calculation, by actual cubing, exactly as in algebra; the only difference being, as regards the conception of the subject, that although i satisfies the equation = 1, it is regarded here as altogether real; namely, as a real right versor* (181).
233. There is no difficulty in conceiving how the same general principles may be extended (comp. 229) to a continued proportion of n+1 complanar vectors,
* This conception differs fundamentally from one which had occurred to several able writers, before the invention of the quaternions; and according to which the symbols 1 and √ − 1 were interpreted as representing a pair of equally long and mutually rectangular right lines, in a given plane. In Quaternions, no line is represented by the number, ONE, except as regards its length; the reason being, mainly, that we require, in the present Calculus, to be able to deal with all possible planes ; and that no one right line is common to all such.