Elements of Quaternions

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Longmans, Green, & Company, 1866 - 762 pages
 

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Contents

CHAPTER II
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On Linear Equations between three Coinitial
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On Plane Geometrical Nets 2024
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higher than the third order but that of R requires the fourth order of differen
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26
28
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On Plane Geometrical Nets resumed 3235
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37
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corresponding to s on the corresponding sheet of the Reciprocal comp
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51
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Among other results of this Chapter a theorem is given in page
43
hat may be called the Spheroidal Eccess of that triangle the total
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CHAPTER III
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refe
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On Quinary Symbols for Points and Planes
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73
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76
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78
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SECTION 30n Amharmonic Coordinates in Space 6267
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82
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86
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90
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92
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Page
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a ſign Fig
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0n Barycentres of Systems of Points and
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circle are + 29 20 and 6 + 30 86 as illustrated by Fig
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with others easily deduced which may all be illustrated by the above
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46
90
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91
53
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of the tico rectangular directions that each such generatrix PP is crossed perpendi
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SECTION 70n Differentials of Vectors 98102
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103
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First Motive for naming the Quotient of
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Many other motives leading to the adoption of the name Quater
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115
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125
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On the Axis and Angle of a Quaternion
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130
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132
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137
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139
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144
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On Radial Quotients and on the Square of
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On WectorArcs and VectorAngles consi
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SECTION 100n a System of Three Right Versors
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SECTION 110n the Tensor of a Vector or of a Quater
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146 215 219 263 287
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y 219 229 267 291
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Pages
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293 174 151 222 234 270
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SECTION 120n the Sum or Difference of any two Qua
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246 27 302 182 158 230 278
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or p p +c a Y or Wap + p WYp+WoWAppl 0 Y
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On the Right Part or Vector Part of
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Section of the second Chapter of the Third Book as III ii 6 and so
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183 159 231 247 279 303 184 161 232 280
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so that by P p xii these three asymptotes compose a real and rect
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may be mentioned
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SECTION 140n the Reduction of the General Quaternion
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On some Geometrical Proofs of the Associative
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ON QUATERNIONS CONSIDERED AS PRODUCTS
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On a Second Method of arriving at the same
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while the three scalars a y z are simply rectangular coordinates from
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162 233 248 281
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of the equation Y2 is expressed by the very simple for
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0n the Fourth Proportional to Three Diplanar
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SECTION 80m an Equivalent Interpretation of the Fourth
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sº 187 166 235 251 283 308
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SECTION 90n a Third Method of interpreting a Product
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a is the vector at the time t of the mass or particle m P is the
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On Powers and Logarithms of Diplanar Qua
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and 3Xm Da? P+ H D
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the independent variable being the are in T while it is arbitrary
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ON DIFFERENTIALS AND DEVELOPMENTS OF FUNCTIONS OF QUA
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SECTION 2Elementary Illustrations of the Definition
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in pp 550561 if we write comp Art 396
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it is ultimately equal p 595 to the quarter of the deviation 397
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Examples of Quaternion Differentiation 409419
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with expressions p 588 for the coefficients or coordinates rº y
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see again 408 e there pass three lines of curvature comp p 677
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The formula p 399 d 41 qdq g
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and u is the small or large angle subtended at the centre K
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SECTION 50n Successive Differentials and Developments
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a 191 170 239 287 311
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ing quaternion functions which vanish together and a form of deve
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another symbol of linear operation which it is shown how to
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hich Y is a rector function of p not generally linear and deduced
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Øs SOME ADDITIONAL APPLICATIONs of QUATERNIoNs witH
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Section 30m Normals and Tangent Planes to Surfaces 501510
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of any central quadric
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SECTION 4On Osculating Planes and Absolute Normals
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so that this new surface is cut by
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curvatures of helix ellipse hyperbola logarithmic spiral
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the function p is said to be selfconjugate then this last cubic has three
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Polar Aris Polar Developable
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fºrences are given to a very interesting Memoir by M de SaintVenant Sur
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face has throughout been arbitrary or general as stated in d
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of the same near point P from the osculating circle at P multiplied
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to a surface Some of the theorems or constructions
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to serve in questions in which sº is neglected are assigned in p 579
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the ution of the surface 1 made by the normal plane to the given
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the entelºpe of a certain variable sphere comp 398 which has
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Osculating Sphere to a curve of double curvature
629
Šiºn 6On Osculating Circles and Spheres to Curves
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a and y any constant values consistent with NI the equation N1
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in SI may be thus decomposed into factors p 666
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foregoing theory for the case of a Central Quadric and especially
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inverse function p + e1 where e is any scalar and thus by chang
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indefinitely many quadrics with a common centre o have their asymp
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spares of the two last semiaxes are the roots may be written p 683
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of which two guar
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surface and R R1 R2 the three corresponding points near to each other
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Surfaces and on Geodetic Curvatures 694698
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selfconjugation is given at a later stage in the few first subarticles to Art 415
698
700
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to facilitate reference e In fact the references in the text of the Elements
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The equation p 704
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and 081 vºdpSvdpdºp W
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whereof the system G contains what may be called the Interme
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so that r and r are unit tangents to the lines of curvature it is easily
719
if dt 0 S d or d ij if dº S
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introducing the two new integrals p 729
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the Sun on the Moon or of one Planet on another which is nearer
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p 735 it is found that the directions of the two forces of the first
736
are rigorously proportional to the numbers 1 and 3 the three forces
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comp the formula W3 in p xlvi by the symbolie and cubic equa
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answering to a given point P thereon may by W1 and X1 be
752
Smith 1853
by the Editor

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