formula, q=ẞ: a; and the versor as the (equal) factor, q, in the converse formula, ß = q.a; where it is still supposed that the two vectors, a and B, are equally long. - 153. A versor, like a radial (147), cannot degenerate into a scalar, except by its angle acquiring one or other of the two limit-values, O and . In the first case, it becomes positive unity; and in the second case, it becomes negative unity: each of these two unit-scalars (147) being here regarded as a factor (or coefficient, comp. 12), which operates on a line, to preserve or to reverse its direction. In this view, we may say that 1 is an Inversor; and that every Right Versor (or versor with an angle is a Semi-inversor:* because it half-inverts the line on which it operates, or turns it through half of two right angles (comp. Fig. 41). For the same reason, we are led to consider every right versor (like every right radial, 149, from which indeed we have just seen, in 152, that it differs only as factor differs from quotient), as being one of the square-roots of negative unity: or as one of the values of the symbol √ −1. 2 154. In fact we may observe that the effect of a right versor, considered as operating on a line (in its own plane), is to turn that line, towards a given hand, through a right angle. If then q be such a versor, and if qa= ß, we shall have also (comp. Fig. 41), =-a; so that, if a be any line in the plane of a right versor q, we have the equation, whence it is natural to write, under the same condition, q2 = − 1, qß = as in 149. On the other hand, no versor, which is not right-angled, can be a value of √-1; or can satisfy the equation q'a = a, as Fig. 42 may serve to illustrate. For it is included in the meaning of this last equation, as applied to the theory of versors, that a rotation through 2q, or through the double of the angle of q itself, is equi This word, "semi-inversor," will not be often used; but the introduction of it here, in passing, seems adapted to throw light on the view taken, in the present work, of the symbol V-1, when regarded as denoting a certain important class (149) of Reals in Geometry. There are uses of that symbol, to denote Geometrical Imaginaries (comp. again Art. 149, and the Notes to page 90), considered as connected with ideal intersections, and with ideal contacts; but with such uses of √ – 1 we have, at present, nothing to do. CHAP. I.] VERSOR OF A QUATERnion, or of a vector. 135 valent to an inversion of direction; and therefore to a rotation through two right angles. 155. In general, if a be any vector, and if a be used as a temporary symbol for the number expressing its length; so that a is here a positive scalar, which bears to positive unity, or to the scalar + 1, the same ratio as that which the length of the line a bears to the assumed unit of length (comp. 128); then the quotient a: a denotes generally (comp. 16) a new vector, which has the same direction as the proposed vector a, but has its length equal to that assumed unit: so that it is (comp. 146) the Unit-Vector in the direction of a. We shall denote this unit-vector by the symbol, Ua; and so shall write, generally, that is, more fully, if a be, as above supposed, the number (commensurable or incommensurable, but positive) which represents that length, with reference to some selected standard. 156. Suppose now that qẞ: a is (as at first) a general quaternion, or the quotient of any two vectors, a and ẞ, whether equal or unequal in length. Such a Quaternion will not (generally) be a Versor (or at least not simply such), according to the definition lately given; because its effect, when operating as a factor (103) on a, will not in general be simply to turn that line (151): but will (generally) alter the length,† as well as the direction. But if we reduce the two proposed vectors, a and ß, to the two unit-vectors Ua and Uẞ (155), and form the quotient of these, we shall then have taken account of relative direction alone: and the result will therefore be a versor, in the sense lately defined (151). We propose to call the quotient, or the versor, thus obtained, the versor-element, or briefly, the VERSOR, of the Quaternion q; and shall find it convenient to em * We shall soon propose a general notation for representing the lengths of vectors, according to which the symbol Ta will denote what has been above called a; but are unwilling to introduce more than one new characteristic of operation, such as K, or T, or U, &c., at one time. By what we shall soon call call an act of tension, which will lead us to the consideration of the tensor of a quaternion. ploy the same Characteristic, U, to denote the operation of taking the versor of a quaternion, as that employed above to denote the operation (155) of reducing a vector to the unit of length, without any change of its direction. On this plan, the symbol Uq will denote the versor of q; and the foregoing definitions will enable us to establish the General Formula : in which the two unit-vectors, Ua and Uß, may be called, by analogy, and for other reasons which will afterwards appear, the versorst of the vectors, a and B. 157. In thus passing from a given quaternion, q, to its versor, Uq, we have only changed (in general) the lengths of the two lines compared, namely, by reducing each to the assumed unit of length (155, 156), without making any change in their directions. Hence the plane (119), the axis (127, 128), and the angle (130), of the quaternion, remain unaltered in this passage; so that we may establish the two following general formulæ : * For the moment, this double use of the characteristic U, to assist in denoting both the unit-vector Ua derived from a given line a, and also the versor Uq derived from a quaternion q, may be regarded as established here by arbitrary definition; but as permitted, because the difference of the symbols, as here a and q, which serve for the present to denote vectors and quaternions, considered as the subjects of these two operations U, will prevent such double use of that characteristic from giving rise to any confusion. But we shall further find that several important analogies are by anticipation expressed, or at least suggested, when the proposed notation is employed. Thus it will be found (comp. the Note to page 119), that every vector a may usefully be equated to that right quotient, of which it is (133) the index; and that then the unit-vector Ua may be, on the same plan, equated to that right radial (147), which is (in the sense lately defined) the versor of that right quotient. We shall also find ourselves led to regard every unit-vector as the axis of a quadrantal (or right) rotation, in a plane perpendicular to that axis; which will supply another inducement, to speak of every such vector as a versor. On the whole, it appears that there will be no inconvenience, but rather a prospective advantage, in our already reading the symbol Ua as versor of a;" just as we may read the analogous symbol Uq, as versor of q." 46 + Compare the Note immediately preceding. CHAP. I.] EQUAL AND RECIPROCAL VERSORS, REVERSORS. 137 Lq=49, and Ax. q = Ax. q, if Uq' = Uq ; the versor of a quaternion depending solely on, but conversely being sufficient to determine, the relative direction (156) of the two lines, of which (as vectors) the quaternion itself is the quotient (112); or the axis and angle of the rotation, in the plane of those two lines, from the divisor to the dividend (128): so that any two quaternions, which have equal versors, must also have equal angles, and equal (or coincident) axes, as is expressed by the last written formula. Conversely, from this dependence of the versor Uq on relative direction alone, it follows that any two quaternions, of which the angles and the axes are equal, have also equal versors; or in symbols, that UqUq, if q'=49, and Ax.q= Ax.q. For example, we saw (in 138) that the conjugate and the reciprocal of any quaternion have thus their angles and their axes the same; it follows, therefore, that the versor of the conjugate is always equal to the versor of the reciprocal; so that we are permitted to establish the following general formula,† it follows that the versor of the reciprocal of any quaternion is, at the same time, the reciprocal of the versor; so that we may write, * The unit-vector Ua, which we have recently proposed (156) to call the versor of the vector a, depends in like manner on the direction of that vector alone; which exclusive reference, in each of these two cases, to DIRECTION, may serve as an additional motive for employing, as we have lately done, one common name, VERSOR, and one common characteristic, U, to assist in describing or denoting both the UnitVector Ua itself, and the Quotient of two such Unit-Vectors, Uq = Uß: Ua; all danger of confusion being sufficiently guarded against (comp. the Note to Art. 156), by the difference of the two symbols, a and q, employed to denote the vector and the quaternion, which are respectively the subjects of the two operations U; while those two operations agree in this essential point, that each serves to eliminate the quantitative element, of absolute or relative length. Compare the Note to Art. 138. T Hence, by the recent result (157), we have also, generally, Also, because the versor Uq is always a radial quotient (151, 152), it is (by 150) the conjugate of its own reciprocal; and therefore at the same time (comp. 145), the reciprocal of its own conjugate; so that the product of two conjugate versors, or what we have called (145, (11.)) their common Norм, is always equal to positive unity; or in symbols (comp. 150), NUq= Uq. KUq = 1. For the same reason, the conjugate of the versor of any quaternion is equal to the reciprocal of that versor, or (by what has just been seen) to the versor of the reciprocal of that quaternion; and therefore also (by 157), to the versor of the conjugate; so that we may write generally, as a summary of recent results, the formula: each of these four symbols denoting a new versor, which has the same plane, and the same angle, as the old or given versor Uq, but has an opposite axis, or an opposite direction of rotation: so that, with respect to that given Versor, it may naturally be called a REVERSOR. 159. As regards the versor itself, whether of a vector or of a quaternion, the definition (155) of Ua gives, Uxa+Ua, or =- Ua, according as x> or < 0; because (by 15) the scalar coefficient x preserves, in the first case, but reverses, in the second case, the direction of the vector a; whence also, by the definition (156) of Uq, we have generally (comp. 126, 143), Uxq= = + Uq, or - Uq, according as a> or < 0. The versor of a scalar, regarded as the limit of a quaternion (131, 139), is equal to positive or negative unity (comp. 147, |