CONTENTS. SECTION 3.-On Anharmonic Co-ordinates in Space, SECTION 4. On Geometrical Nets in Space, SECTION 5. On Barycentres of Systems of Points; and on Simple and Complex Means of Vectors, SECTION 6. On Anharmonic Equations, and Vector Expressions, of Surfaces and Curves in Space, SECTION 7.-On Differentials of Vectors, An application of finite differences, to a question connected with barycentres, occurs in p. 87. The anharmonic generation of a ruled hyperboloid (or paraboloid) is employed to illustrate anharmonic equations; and (among other examples) certain cones, of the second and third orders, have their vector equations assigned. In the last Section, a definition of differentials (of vectors and scalars) is proposed, which is afterwards extended to differentials of quaternions, and which is independent of developments and of infinitesimals, but involves the conception of limits. Vectors of Velocity and Acceleration are mentioned; and a hint of Hodographs is given. BOOK II. ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF . 111 Pages. 62-67 67-85 85-89 90-97 98-102 103-300 CHAPTER I. FUNDAMENTAL PRINCIPLES RESPECTING QUOTIENTS OF VECTORS, 103-239 Very little, if any, of this Chapter II. i., should be omitted, even in a first perusal; since it contains the most essential conceptions and notations of the Calculus of Quaternions, at least so far as quotients of vectors are concerned, with numerous geometrical illustrations. Still there are a few investigations respecting circumscribed cones, imaginary intersections, and ellipsoids, in the thirteenth Section, which a student may pass over, and which will be indicated in the proper place in this Table. SECTION 1. Introductory Remarks; First Principles adopted from Algebra, 103-106 SECTION 2.-First Motive for naming the Quotient of two 106-110 SECTION 3.-Additional Illustrations, 110-112 It is shown, by consideration of an angle on a desk, or inclined plane, that the complex relation of one vector to another, in length and Pages. in direction, involves generally a system of four numerical elements. Many other motives, leading to the adoption of the name, "Quaternion," for the subject of the present Calculus, from its fundamental connexion with the number "Four," are found to present themselves in the course of the work. SECTION 4. On Equality of Quaternions; and on the Plane of a Quaternion,. .. 112-117 117-120 SECTION 5. On the Axis and Angle of a Quaternion; and Quaternion, 129-133 SECTION 8. On the Versor of a Quaternion, or of a Vector; and on some General Formulæ of Transformation, 133-142 In the five foregoing Sections it is shown, among other things, that the plane of a quaternion is generally an essential element of its constitution, so that diplanar quaternions are unequal; but that the square of every right radial (or right versor) is equal to negative unity, whatever its plane may be. The Symbol V - 1 admits then of a real interpretation, in this as in several other systems; but when thus treated as real, it is in the present Calculus too vague to be useful: on which account it is found convenient to retain the old signification of that symbol, as denoting the (uninterpreted) Imaginary of Algebra, or what may here be called the scalar imaginary, in investigations respecting non-real intersections, or non-real contacts, in geometry. 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, This Section is important, on account of its constructions of multiplication and division; which show that the product of two diplanar versors, and therefore of two such quaternions, is not independent of the order of the factors. SECTION 10. On a System of Three Right Versors, in The student ought to make himself familiar with these laws, which are all included in the Fundamental Formula, 142-157 157-162 Pages. (B) In fact, a QUATERNION may be symbolically defined to be a Quadrinomial Expression of the form, q=w+ix+jy+kz, in which w, z, y, z are four scalars, or ordinary algebraic quantities, while i, j, k are three new symbols, obeying the laws contained in the formula (A), and therefore not subject to all the usual rules of algebra: since we have, for instance, y=+k, but ji=-k; and jド = - (ijk)2. SECTION 11. On the Tensor of a Vector, or of a Quaternion; and on the Product or Quotient of any two Quaternions, SECTION 12. On the Sum or Difference of any two Quaternions; and on the Scalar (or Scalar Part) of a Quaternion, . SECTION 13. On the Right Part (or Vector Part) of a SECTION 14. On the Reduction of the General Quaternion Articles 213-220 (with their sub-articles), in pp. 214-233, may be omitted at first reading. 162-174 175-190 190-233 233-239 CHAPTER II. ON COMPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN The first six Sections of this Chapter (II. ii.) may be passed over in a first perusal. SECTION 1.-On Complanar Proportion of Vectors; Fourth Proportional to Three, Third Proportional to Two, Mean Proportional, Square Root; General Reduction of a Quaternion in a given Plane, to a Standard Binomial Form, SECTION 2.-On Continued Proportion of Four or more Vectors; whole Powers and Roots of Quaternions; and Roots of Unity, 240-285 240-246 246-251 Pages. SECTION 3.-On the Amplitudes of Quaternions in a given SECTION 4. On the Ponential and Logarithm of a Quater- SECTION 5.-On Finite (or Polynomial) Equations of Alge- SECTION 6.- On the n2 n Imaginary (or Symbolical) SECTION 7. On the Reciprocal of a Vector, and on Har- In this last Section (II. ii. 7) the short first Article 258, and the following Art. 259, as far as the formula VIII. in p. 280, should be read, as a preparation for the Third Book, to which the Student may next proceed. 251-257 257-264 265-275 275-279 279-285 CHAPTER III. ON DIPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN L This Chapter may be omitted, in a first perusal. SECTION 1. On some Enunciations of the Associative Property, or Principle, of Multiplication of Diplanar Quaternions, 286-300 286-293 SECTION 2.- On some Geometrical Proofs of the Associative Property of Multiplication of Quaternions, which are independent of the Distributive Principle, 293-297 SECTION 3. On some Additional Formulæ, 297-300 ON QUATERNIONS, CONSIDERED AS PRODUCTS OR CHAPTER I. Pages. 301 to the end. ON THE INTERPRETATION OF A PRODUCT OF VECTORS, OR POWER . The first six Sections of this Chapter ought to be read, even in a first perusal of the work. SECTION 1. On a First Method of Interpreting a Product of Two Vectors as a Quaternion, SECTION 2.-On some Consequences of the foregoing Interpretation, This first interpretation treats the product β. a, as equal to the quotient 3: a-1; where a ̄1 (or Ra) is the previously defined Reciprocal (II. ii. 7) of the vector a, namely a second vector, which has an inverse length, and an opposite direction. Multiplication of Vectors is thus proved to be (like that of Quaternions) a Distributive, but not generally a Commutative Operation. The Square of a Vector is shown to be always a Negative Scalar, namely the negative of the square of the tensor of that vector, or of the number which expresses its length ; and some geometrical applications of this fertile principle, to spheres, &c., are given. The Index of the Right Part of a Product of Two Coinitial Vectors, Οа, ов, is proved to be a right line, perpendicular to the Plane of the Triangle oab, and representing by its length the Double Area of that triangle; while the Rotation round this Index, from the Multiplier to the Multiplicand, is positive. This right part, or vector part, Vaß, of the product vanishes, when the factors are parallel (to one common line); and the scalar part, Saß, when they are rectangular. 301-390 301-303 303-308 SECTION 3. On a Second Method of arriving at the same 308-310 SECTION 4. On the Symbolical Identification of a Right 310-313 SECTION 5. On some Simplifications of Notation, or of Expression, resulting from this Identification; and on the Conception of an Unit-Line as a Right Versor, 313-316 |