By Arthur S. Hathaway
Illustrated, together with various 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 directly 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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Additional resources for A primer of quaternions - illustrated
Hence the theorem. Note. This theorem shows that a number extends and turns vectors into vectors in the same way that it extends and turns steps into steps. Moreover, when the vector is not perpendicular to the axis of the multiplier, there is no resulting vector, since in the case of the corresponding step there is no resulting step. In the case of a vector multiplicand, that is oblique to the axis of q, the product is an actual quaternion that is not a vector, while in the case of the corresponding step multiplicand the product belongs to that class of products in which the multiplicand does not admit of the operation of the multiplier, as √ in 2 universities, -2 countries, etc.
A self-conjugate strain has by Art. 88 three mutually perpendicular latent directions, and conversely, if φ have three mutually perpendicular latent directions, i, j, k, corresponding to latent roots a, b, c, then φρ = −aiSiρ − bjSjρ − ckSkρ, which is self-conjugate. ] 92. 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.
CHAPTER 4. EQUATIONS OF FIRST DEGREE 50 75. A plane of particles parallel to α, β is homogeneously spread and turned by the strain φ into a plane of particles parallel to φα, φβ, and the ratio of extension and turning is V φαφβ/V αβ. For let AP Q be a plane parallel to α, β, and let A, P , Q strain into A , P , Q . Then, since AP = xα + yβ, AQ = x α + y β, therefore A P = xφα + yφβ, A Q = x φα + y φβ. 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.
A primer of quaternions - illustrated by Arthur S. Hathaway