A Primer Of Quaternions - Illustrated by Arthur S. Hathaway

By Arthur S. Hathaway

Illustrated, together with a number of Examples - Chapters: Definitions And Theorems - heart 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 traces And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so on.

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Chapter 4 Equations of First Degree 61. The general equation of first degree in an unknown vector ρ is of the form, (a) q1 ρr1 + q2 ρr2 + · · · = q, where q, q1 , r1 , q2 , r2 , · · · are known numbers. This equation may be resolved into two equations by taking the scalar and the vector of each member; and we shall consider these equations separately. 62. Taking the scalar of (a), Art. 61, the term Sq1 ρr1 becomes, by a cyclic change in the factors, S · r1 q1 ρ, and this becomes [by dropping the vector (Sr1 q1 )ρ, since its scalar is zero] S(V r1 q1 · ρ); and similarly for the other terms.

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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.

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