An Elementary Treatise on QuaternionsAn Elementary Treatise on Quaternions by Peter Guthrie Tait, first published in 1890, is a rare manuscript, the original residing in one of the great libraries of the world. This book is a reproduction of that original, which has been scanned and cleaned by state-of-the-art publishing tools for better readability and enhanced appreciation. Restoration Editors' mission is to bring long out of print manuscripts back to life. Some smudges, annotations or unclear text may still exist, due to permanent damage to the original work. We believe the literary significance of the text justifies offering this reproduction, allowing a new generation to appreciate it. |
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Common terms and phrases
a₁ arcs axis B₁ Cartesian centre Chapter circle commutative law cone conjugate constant coplanar curve diameters differential direction drawn easily ellipsoid envelop equal equivalent evidently expression Find the equation Find the locus formula geometrical given equation given lines given point given vectors gives Hamilton Hence indeterminate intersection inverse function LAOB last section length linear and vector multiply obviously origin osculating plane P₁ parallel perpendicular properties proved quaternion radius rectangular represented right angles rotation S.aßy Saß scalar equations second order self-conjugate sides solution sphere spherical conic spherical excess ẞ² straight line student surface surface of revolution tangent plane Taylor's Theorem tensor theorem three vectors triangle unit-vector Vaß vector function vector perpendicular versor written Γλμ φρ
Popular passages
Page 174 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Page 149 - Find the locus of a point the sum of the squares of whose distances from two given points is constant.
Page 217 - To find the locus of the foot of the perpendicular drawn from the origin to a tangent plane to any surface.
Page 150 - Sftp = 0, Syp = 0. 20. If three mutually perpendicular vectors be drawn from a point to a plane, the sum of the reciprocals of the squares of their lengths is independent of their directions. 21. Find the general form of the equation of a plane from the condition (which is to be assumed as a definition) that any two planes intersect in a single straight line. CHAPTER VII THE SPHERE AND CYCLIC CONE.
Page 3 - What reason do writers on quaternions give for taking xx' + yy' + zz' negatively in the case of the product of two vectors? In the passage quoted above Professor Tail refers to section 9 of his Treatise for the proof that the square of a unit vector is — 1. There we find: "It may be interesting, at this stage, to anticipate so far as to remark that in the theory of quaternions the analogue of cos 6 + j/ — 1 sin d is cos (I + at sin 0, where w
Page 23 - Then, generally, p may be expressed as the sum of a number of terms, each of which is a...
Page 267 - S.vdp = 0 ................ (2), v is a vector perpendicular to the surface, and its length is inversely proportional to the normal distance between two consecutive surfaces. In fact (2) shows that v is perpendicular to dp, which is any tangent vector, thus proving the first assertion. Also, since in passing to a proximate surface we may write we see that F (p + v~l SC) = C + SC.
Page 195 - Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane.
Page 30 - Obtain the expression for the general value ot all angles whose 3. Obtain cos (A — B) in terms of sines and cosines of A and B ; and cos A — cos B as a product of sines or cosines. 4. Prove that sin -^-= +s/Ti-sm A + ^/l— sin A, and determine which signs are to be used when A is between 270° and 360°. /•x , A sin A 5. Prove (i) tan- = •2 1+cosJ
Page 175 - ... that is, a plane parallel to the tangent plane at the point where OA cuts the surface. And (d.) shows that this relation is reciprocal — so that if /3 be any value of...