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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,
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.
ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF
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, .
SECTION 2.-First Motive for naming the Quotient of two
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
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,.
SECTION 5. On the Axis and Angle of a Quaternion; and on the Index of a Right Quotient, or Quaternion, . . 117-120 SECTION 6.-On the Reciprocal, Conjugate, Opposite, and
Norm of a Quaternion; and on Null Quaternions, 120-129 SECTION 7.-On Radial Quotients; and on the Square of a Quaternion, . 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,
In fact, a QUATERNION may be symbolically defined to be a Quadrinomisi Expression of the form,
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,
ÿj=+k, but jï=-k; and k2 —— (ÿjk)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 to a Standard Quadrinomial Form; with a First Proof of the Associative Principle of Multiplication of Quaternions,
Articles 213–220 (with their sub-articles), in pp. 214–233, may be omitted at first reading.
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
SECTION 2.-On Continued Proportion of Four or more Vectors; whole Powers and Roots of Quaternions; and
Roots of Unity,
SECTION 3.-On the Amplitudes of Quaternions in a given
SECTION 6. On the n2n Imaginary (or Symbolical)
SECTION 7.-On the Reciprocal of a Vector, and on Harmonic Means of Vectors; with Remarks on the Anharmonic Quaternion of a Group of Four Points, and on Conditions of Concircularity,
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.
ON DIPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN
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,
SECTION 2.-On some Geometrical Proofs of the Associative Property of Multiplication of Quaternions, which are independent of the Distributive Principle,
SECTION 3.-On some Additional Formulæ,
ON QUATERNIONS, CONSIDERED AS PRODUCTS OR
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
This first interpretation treats the product ß. a, as equal to the quotient 3: a-1; where a1 (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, OA, OB, 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.
SECTION 3.-On a Second Method of arriving at the same