A primer of quaternions - illustrated by Arthur S. Hathaway

By Arthur S. Hathaway

Illustrated, together with quite a few Examples - Chapters: Definitions And Theorems - middle Of Gravity - Curve Tracing, Tangents - Parallel Projection - Step Projection - Definitions And Theorems Of Rotation - Definitions Of flip And Arc Steps - Quaternions - Powers And Roots - illustration Of Vectors - formulation - Equations Of First measure - Scalar Equations, aircraft And immediately Line - Nonions - Linear Homogeneous pressure - Finite And Null traces - Derived Moduli, Latent Roots - Latent traces And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so forth.

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Hence this product multiplies out as if φ were a scalar, and is φ3 − (g1 + g2 + g3 )φ2 + (g2 g3 + g3 g1 + g1 g2 )φ − g1 g2 g3 . CHAPTER 4. EQUATIONS OF FIRST DEGREE 49 Linear Homogeneous Strain 73. , that are determined by OA = φOA, OB = φOB, OC = φOC, etc. In general, any particle P whose vector is OP = ρ occupies after the strain the position P , whose vector is OP = φρ. The particle at O is not moved, since its vector after strain is φOO = φ0 = 0. (a) We have, also, φAP = A P , etc. For, A P = OP − OA = φOP − φOA = φ(OP − OA) = φAP, etc.

So √ −1 φ = χR, where χ = RψR = φφ . 23. Show that φ · V φβφγ = V βγ · mod φ. ] 24. Show that the strain φρ = ρ−aαSβρ, where α, β, are perpendicular unit vectors, consists of a shearing of all planes perpendicular to β, the amount and direction of sliding of each plane being aα per unit distance of the plane from O. 25. Determine ψ and R of Ex. 22 for the strain of Ex. 24, and find the latent directions and roots of ψ. Chapter 5 PROJECT GUTENBERG ”SMALL PRINT” End of The Project Gutenberg EBook of A Primer of Quaternions, by Arthur S.

A singly null nonion strains each line in its null direction into a definite point of its plane; and a doubly null nonion strains each plane that is parallel to its null plane into a definite point of its line. , they are particles of a line parallel to α. So, if φ is doubly null, say CHAPTER 4. , they are particles of a plane parallel to α, β. —It follows similarly that the strain φ alters the dimensions of a line, plane, or volume by as many dimensions as the substance strained contains independent null directions of φ, and no more.

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