Elements of Quaternions - Hardcover

ELEMENTS OF QUATERNIONS by A. S. HARDY. Originally published in 1891. PREFACE: object of the following treatise is to exhibit the elementary principles and notation of the Quaternion Calculus, so as to meet the wants of beginners in the class-room. The Elements and Lectures of Sir William Rowan Hamilton, while they may be said to contain the suggestion of all that will be done in the way of Quaternion research and application, are not, for this reason, as also on account of their diffuseness of style, suitable for the purposes of elementary instruction. Taits work on Quaternions is also, in its originality and conciseness, beyond the time and needs of the beginner. In addition to the above, the following works have been consulted Calcolo dei Quaternione. Bellavitis Modena, 1858. Exposition de la MetTiode des fiquipollences Traduit de 1 Italien de Giusto Bellavitis, par C.-A. Laisant Paris, 1874. Original memoir in the Memoirs of the Italian Society. 1854. Thorie lementaire des Quantites Complexes. J. Hoiiel Paris, 1874. Essai sur une Manure de Representer les Quantites Imaginaires dans les Construction G-eometriques. Par R. Argand Paris, 1806. Second edition, with preface. by J. Hoiiel Paris, 1874. Translated, with notes, from the French, by A. S. Hardy. Van Nostrands Science Series, No. 52 1881. Kurze Anleitunff zum Rechnen mit den Hamilton sclieri Quaternionen. J. Odstrcil Halle, 18T9. Applications Mecaniques du Qalcul des Quaternions. Laisant Paris, 1877. Introduction to Quaternions. Kelland and Tait Lon don, 1873. A free use has been made of the examples and exercises of the last work and, in Article 87, is given, by permis sion, the substance of a paper from Volume L, page 379, American Journal of Mathematics, illustrating admirably the simplicity and brevity of the Quaternion method. If this presentation of the principles shall afford the undergraduate student a glimpse of this elegant and pow erful instrument of analytical research, or lead him to follow their more extended application in the works above cited, the aim of this treatise will have been accomplished. The author expresses his obligation to Mr. T. W. D. Worthen for valuable assistance in the preparation of this work, and to Mr, J. S. Gushing for whatever of typographical excellence it possesses. A. S. HABDY. HANOVER, N. H., June 21, 1881. Contents include: CHAPTER I. Addition and Subtraction of Vectors or, Geometric Addition and Subtraction. Article Paga 1. Definition of a vector. Effect of the minus sign before a vector 1 2. Equal vectors 2 3. Unequal vectors. Vector addition 2 4. Vector addition, commutative 3 5. Vector addition, associative 3 6. Transposition of terms in a vector equation 4 7. Definition of a tensor 4 8. Definition of a scalar 5 9. Distributive law in the multiplication of vector by scalar quantities 6 10. If Sa S 3 0, then 2a and 2 5 7 11. Examples 8 12. Complanar vectors Condition of complanarity 15 13. Co-initial vectors Condition of collinearity 16 14. Examples 17 15. Expression for a medial vector 24 16. Expression for an angle-bisector 25 17. Examples 26 18. Mean point 28 19. Examples 28 20. Exercises 30 CHAPTER II. Multiplication and Division of Vectors or, Geometric Multiplication and Division. 21. Elements of a quaternion 32 22. Equal quaternions 34 28. Positive rotation 35 VI. Article Page 24 Analytical expression for a quaternion. Product and quo tient of rectangular unit-vectors. Tensor and versor of a quaternion ............... ....... 36 25 Symbolic notation q TqVq ................ 39 26. Reciprocal of a quaternion ................ 39 27. Quadrantal versors, i j, ft ................ 40 28. Whole powers of unit vectors. Square of a unit vector is 1, 41 29...