CHAP. I.] GENERAL TRANSFORMATIONS of a versor. 139 153), according as the scalar itself is positive or negative; or in symbols, Ux+1, or = 1, according as = x> or <0; the plane and axis of each of these two unit scalars (147), considered as versors (153), being (as we have already seen) indeterminate. The versor of a null quaternion (141) must be regarded as wholly arbitrary, unless we happen to know a law,* according to which the quaternion tends to zero, before actually reaching that limit; in which latter case, the plane, the axis, and the angle of the versort U0 may all become determined, as limits deduced from that law. The versor of a right quotient (132), or of a right-angled quaternion (141), is always a right radial (147), or a right versor (153); and therefore is, as such, one of the square roots of negative unity (149), or one of the values of the symbol √ −1; while (by 150) the axis and the index of such a versor coincide; and in like manner its reciprocal, its conjugate, and its opposite are all equal to each other. 160. It is evident that if a proposed quaternion q be already a versor (151), in the sense of being a radial (146), the operation of taking its versor (156) produces no change; and in like manner that, if a given vector a be already an unit-vector, it remains the same vector, when it is divided (155) by its own length; that is, in this case, by the number one. For example, we have assumed (128, 129), that the axis of every quaternion is an unit-vector; we may therefore write, generally, in the notation of 155, the equation, U (Ax. g) = Ax.q. A second operation U leaves thus the result of the first operation U unchanged, whether the subject of such successive operations be a line, or a quaternion; we have therefore the two Compare the Note to Art. 131. + When the zero in this symbol, U0, is considered as denoting a null vector (2), the symbol itself denotes generally, by the foregoing principles, an indeterminate unit-vector; although the direction of this unit-vector may, in certain questions, bedetermined, as a limit resulting from a law. come following general formulæ, differing only in the symbols of that subject: UUa = Ua; UUq = Uq; whence, by abstracting (comp. 145) from the subject of the operation, we may write, briefly and symbolically, UUUU. 161. Hence, with the help of 145, 158, 159, we easily deduce the following (among other) transformations of the versor of a quaternion: 1 U 1 1 Kq = = Kq=KUg = (Uq)2= U (q2) = Uq2 ; the parentheses being here unnecessary, because (as will soon be more fully seen) the symbol Uq' denotes one common versor, whether we interpret it as denoting the square of the versor, or as the versor of the square, of q. The present Calculus will be found to abound in General Transformations of this sort; which all (or nearly all), like the foregoing, depend ultimately on very simple geometrical conceptions; but which, notwithstanding (or rather, perhaps, on account of) this extreme simplicity of their origin, are often useful, as elements of a new kind of Symbolical Language in Geometry: and generally, as instruments of expression, in all those mathematical or physical researches to which the Calculus of Quaternions can be applied. It is, however, by no means necessary that a student of the subject, at the present stage, should make himself familiar with all the recent transformations of Uq; although it may be well that he should satisfy himself of their correctness, in doing which the following remarks will perhaps be found to assist. (1.) To give a geometrical illustration, which may also serve as a proof, of the recent equation, q : Kq = (Uq)2, CHAP. 1.] GEOMETRICAL ILLUSTRATIONS. 16 we may employ Fig. 36, bis; in which, by 145, (2.), we have 141 ОА C (2.) As regards the equation, U(q2) = (Uq)2, we have only to conceive that the three lines OA, OB, OC, of Fig. 42, are cut (as in Fig. 42, bis) in three new points, A', B', c', by an unit-circle (or by a circle with a radius equal to the unit of length), which is described about their common origin o as centre, and in their common plane; for then if these three lines be called a, ß, y, the three new lines oa', OB', oc' are (by 155) the three unit-vectors denoted by the symbols, La, Uẞ, Uy; and we have the transformations (comp. 148, 149), UYUY = a Τα OA (3.) As regards other recent transformations (161), although we have seen (135) that it is not necessary to invent any new or peculiar symbol, to represent the reciprocal of a quaternion, yet if, for the sake of present convenience, and as a merely temporary notation, we write Α' Λ Fig. 42, bis. employing thus, for a moment, the letter R as a characteristic of reciprocation, or of the operation of taking the reciprocal, we shall then have the symbolical equations (comp. 145, 158): R2 = K2 = 1; RK = KR; KU URKU = UK; = but we have also (by 160), U2= U; whence it easily follows that U=RUR=RKU=RUK = KUR = KRU = KUK = expresses that the locus of the point P is the indefinite right line, or ray (comp. 132, (4.)), which is drawn from o in the direction of OB,* but not in the opposite direction; because it is equivalent to expresses (comp. 132, (5.)) that the locus of P is the opposite ray from 0; or that it is the indefinite prolongation of the revector BO; because it may be transformed to * In 132, (4.), p. 119, o▲ and a ought to have been OB and B. (6.) If a, ß, y denote (as in sub-art. 2) the three lines OA, OB, OC of Fig. 42 (or of Fig. 42, bis), so that (by 149) we have the equation = (2)*, α α then this other expresses generally that the locus of P is the system of the two last loci; or that it is the whole indefinite right line, both ways prolonged, through the two points o and B (comp. 144, (2.)). (7.) But if it happen that the line y, or oc, like oa′ in Fig. 41 (or in Fig. 41, bis), has the direction opposite to that of a, or of OA, so that the last equation takes the particular form, then U must be (by 154) a right versor; and reciprocally, every right versor, with a plane containing a, will be (by 153) a value satisfying the equation. In this case, therefore, the locus of the point P is (as in 132, (1.), or in 144, (1.)) the plane through o, perpendicular to the line OA; and the recent equation itself, if supposed to be satified by a real* vector p, may be put under either of these two earlier but equivalent forms: SECTION 9.-On Vector-Arcs, and Vector-Angles, considered as Representatives of Versors of Quaternions; and on the Multiplication and Division of any one such Versor by another. 162. Since every unit-vector oa (129), drawn from the origin o, terminates in some point A on the surface of what we have called the unit-sphere (128), that term a (1) may be considered as a Representative Point, of which the position on that surface determines, and may be said to represent, the direction of the line os in space; or of that line multiplied (12, 17) by any positive scalar. And then the Quaternion which is the quotient (112) of any two such unit-vectors, and which is in one view a Radial (146), and in another view a Versor (151), may be said to have the arc of a great circle, AB, upon the unit sphere, which connects the terms of the two # Compare 149, (2.); also the second Note to the same Article; and the Notes to page 90. CHAP. 1.] REPRESENTATIVE AND VECTOR ARCS. 143 vectors, for its Representative Arc. We may also call this arc a VECTOR ARC, on account of its having a definite direction (comp. Art. 1), such as is indicated (for example) by a curved arrow in Fig. 39 and as being thus contrasted with its own opposite, or with what may be called by analogy the Revector Arc BA (comp. again 1): this latter arc representing, on the present plan, at once the reciprocal (134), and the conjugate (137), of the former versor; because it represents the corresponding Reversor (158). 163. This mode of representation, of versors of quaternions by vector arcs, would obviously be very imperfect, unless equals were to be represented by equals. We shall therefore define, as it is otherwise natural to do, that a vector arc, AB, upon the unit sphere, is equal to every other vector arc CD which can be derived from it, by simply causing (or conceiving) it to slide in its own great circle, without any change of length, or reversal of direction. In fact, the two isosceles and plane triangles AOB, COD, which have the origin o for their common vector, and rest upon the chords of these two arcs as bases, are thus complanar, similar, and similarly turned; so that (by 117, 118) we may here write, the condition of the equality of the quotients (that is, here, of the versors), represented by the two arcs, being thus satisfied. We shall sometimes denote this sort of equality of two vector arcs, AB and CD, by the formula, ~ AB = CD; and then it is clear (comp. 125, and the earlier Art. 3) that we shall also have, by what may be called inversion and alternation, these two other formulæ of arcual equality, o D B A Fig. 35, bis. * Some aid to the conception may here be derived from the inspection of Fig 34; in which two equal angles are supposed to be traced on the surface of one com |