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, airplane And directly Line - Nonions - Linear Homogeneous pressure - Finite And Null lines - Derived Moduli, Latent Roots - Latent traces And Planes - Conjugate Nonions - Self-Conjugate Nonions - Etc., and so on.

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Cor. 1. The conjugate of a vector is the negative vector. [39 Cor. ] • Cor. 2. The conjugate of a product of two vectors is the product of the same vectors in reverse order. [Art. 39, Cor. ] • Cor. 3. 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.

Sσ(φψ) ρ = Sρφψσ = Sρφ(ψσ) = Sψσφ ρ = Sσψ φ ρ, and therefore (φψ) = ψ φ . [If Sσ(α − β) = 0 for all values of σ, then α − β = 0, since no vector is perpendicular to every vector σ. ] 89. Two conjugate strains have the same latent roots and moduli, and a latent plane of one is perpendicular to the corresponding latent line of the other. For since (φ − g1 )α = 0, therefore 0 = Sρ(φ − g1 )α = Sα(φ − g1 )ρ, and therefore φ − g1 is a null nonion whose plane is perpendicular to α. Hence g1 is a latent root of φ , and the latent plane of φ corresponding to φ − g1 is perpendicular to the latent line of φ corresponding to φ − g1 .

10. Show that pq rotates into qp, and determine two such rotations. 11. Show that SKq = Sq, V Kq = −V q. 12. Show that Kαβ = βα, Sαβ = Sβα, V αβ = −V βα. 13. Show that Kαβγ = −γβα; V αβγ = V γβα; Sαβγ = Sβγα = Sγαβ = −Sγβα = −Sβαγ = −Sαγβ. (a) Determine the conjugate of a product of n vectors. 14. Prove by diagram that Kpq = Kq · Kp. Addition 47. Definition. The sum (p + q) is the number determined by the condition that its product is the sum of the products of p and q. CHAPTER 3. QUATERNIONS 32 Thus let OA be any step that is multiplied by both p and q, and let pOA = OB, qOA = OC, and OB + OC = OD, then (p + q)OA = OD.