By Arthur S. Hathaway
Illustrated, together with a number of 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 strains And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so on.
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Additional info for A Primer of Quaternions - Illustrated
68. The point of intersection of the three planes Sαρ = a, Sβρ = b, Sγρ = c is [Art. ] ρ= (aV βγ + bV γα + cV αβ) . Sαβγ Examples 1. Find the equation of the locus of a point that moves so that its numerical distances from two fixed points are equal. 2. A point moves so that its scalar distances from two fixed planes are equal; show that its locus is a plane bisector of the diedral angle of the given planes. 3. A point moves so that the sum or difference of its scalar distances from two fixed planes is constant; show that its locus is a plane parallel to the interior or exterior bisector of the diedral angle of the given planes.
If p, q, r be any quaternions, then (p + q)r = pr + qr. For we have, (1 + qp−1 )p · r = (1 + qp−1 ) · pr, and expanding each member by the preceding lemma, we have, (p + q)r = pr + qr. , (p + q + r + s)t = [(p + q) + (r + s)]t = (p + q)t + (r + s)t = pt + qt + rt + st. • Cor. 1. r(p + q) = rp + rq. For let p , q , r be the conjugates of p, q, r. Then from (p + q )r = p r + q r , we have, by taking the conjugates of each member, r(p + q) = rp + rq. [Art. 39, Cor. 2; Art. ] • Cor. 2. A product of sums equals the sum of all partial products that may be formed from the given product by multiplying together, in the order in which they stand, a term from each factor of the product.
Since a cubic with real coefficients has at least one real root, therefore a real nonion has at least one latent direction. Also if two roots are imaginary, they are conjugate imaginaries, and the corresponding latent directions must also be conjugate imaginaries. 85. If α, β, γ be the latent directions corresponding to g1 , g2 , g3 , then (β, γ), (γ, α), (α, β) determine latent planes of φ in which the ratios of spreading are g2 g3 , g3 g1 , g1 g2 . , V φβφγ = V (g2 β · g3 γ) = g2 g3 V βγ. Hence, in the general case when the latent roots are all unequal, the latent vectors α, β, γ must form a non-coplanar system, since any two of the latent lines or planes determined by them have unequal ratios of extension, and cannot, therefore, coincide.
A Primer of Quaternions - Illustrated by Arthur S. Hathaway