By Arthur S. Hathaway
Illustrated, together with various 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, airplane And immediately Line - Nonions - Linear Homogeneous pressure - Finite And Null traces - Derived Moduli, Latent Roots - Latent strains And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so forth.
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Extra resources for A Primer of Quaternions - Illustrated
The scalar measure of the projection of β upon α is −Sαβ Tα , and the tensor measure of the projection of β perpendicular to α is T V αβ T α . ] • Cor. 3. If θ be the angle between α, β, then cos θ = − TSαβ αβ , sin θ = T V αβ T αβ . ] 60. The volume of a parallelepiped on α, β, γ as edges is −Sαβγ (the volume being positive or negative according as α lies on the counter-clockwise or clockwise side of β, γ). For let α1 be the projection of α upon V βγ; then taking the face β, γ of the parallelepiped as the base, we have by Art.
A real self-conjugate strain has real latent roots. √ √ For let α = α + β −1, β = α − β −1 be latent directions corresponding to conjugate imaginary roots a, b of a real nonion φ; then, if φ is selfconjugate, we have Sα φβ = Sβ φα = bSα β = aSα β , or, since a, b are unequal, therefore Sα β = 0; but this is impossible, since Sα β = α2 + β 2 , a negative quantity. Therefore φ is not self-conjugate if it has imaginary latent roots. 93. A nonion φ is negatively self-conjugate when φ = −φ, or when Sσφρ = −Sρφσ.
The conjugate of a product of three vectors is the negative of the product of the same vectors in reverse order. [Art. 39, Cor. ] CHAPTER 3. QUATERNIONS 27 The Rotator q()q −1 41. , OB/OA determines the number r such that rOA = OB. By Art. 21, equal step ratios determine equal numbers. If the several pairs of steps that are in a given ratio r be given a rotation whose equatorial arc is 2 arc q, they are still equal ratios in their new positions and determine a new number r that is called the number r rotated through 2 arc q.
A Primer of Quaternions - Illustrated by Arthur S. Hathaway