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 info for A Primer of Quaternions - Illustrated
By Arts. 59, 55, (e), the directed area of the triangle AP Q is 21 V · AP · AQ = 12 (xy − x y)V αβ, and the directed area of the triangle A P Q is the same multiple of V φαφβ. Hence the ratio of the extension and turning of directed area is V φαφβ/V αβ. 76. A volume of particles is homogeneously dilated by the strain φ in the ratio Sφαφβφγ/Sαβγ, where α, β, γ are any given non-coplanar vectors. For let the pyramid AP QR strain into the pyramid A P Q R . Then since AP = xα + yβ + zγ, AQ = x α + y β + z γ, AR = x α + y β + z γ, therefore A P , A Q , A R have these values with φα, φβ, φγ instead of α, β, γ.
Show that the nonion of Ex. 16 takes the form bρ − f (αSβρ + βSαρ), where b is the mean latent root. 19. Substitute the nonion of Ex. 18 for φ in Ex. 6 and show that the quadric surface is cut in circles by planes perpendicular to α or β. When is the surface one of revolution? 20. If the conjugate of a nonion is its reciprocal, and the modulus is positive, then the nonion is a rotation; and conversely every rotation satisfies this condition. , T Rρ/ρ = 1. Also Sρσ = SρR−1 Rσ = SRρRσ and therefore the angle between ρ, σ = ∠ between Rρ, Rσ.
The product of a tensor and a versor is a number with that tensor and versor; and conversely, a number is the product of its tensor and its versor. For if n be a tensor, and q a versor, then nq turns by the factor q and extends by the factor n, and vice versa for q n; hence either of the products, nq , q n, is a quaternion with tensor n and versor q . Similarly, q = T q · U q = U q · T q. 38. Any successive factors of a product may be replaced by their product without altering the value of the whole product; but in general such factors can be changed in order without altering the value of the product only when those factors are cocircular.
A Primer of Quaternions - Illustrated by Arthur S. Hathaway