An Elementary Treatise on Quaternions, |
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Page vi
... results , and of elegant yet powerful analytical investigations , such as are contained in the writings of but a very few of the greatest mathematicians . For a succinct account of the steps by which Hamilton was led to the invention of ...
... results , and of elegant yet powerful analytical investigations , such as are contained in the writings of but a very few of the greatest mathematicians . For a succinct account of the steps by which Hamilton was led to the invention of ...
Page xvi
... . Geometrical interpretation of these results , §§ 162–166 . Proof of the transformation pp = pp + q V. ( i + ck ) p ( i — ck ) where ( 4-9 , ) i = 0 , ( 4 - g , ) k = 0 , e2 = 92-93 , 91-92 p = ( 91 + xvi CONTENTS .
... . Geometrical interpretation of these results , §§ 162–166 . Proof of the transformation pp = pp + q V. ( i + ck ) p ( i — ck ) where ( 4-9 , ) i = 0 , ( 4 - g , ) k = 0 , e2 = 92-93 , 91-92 p = ( 91 + xvi CONTENTS .
Page 4
... results of Wallis and De Moivre . They attempted to express as a line the product of two lines each represented by a symbol such as a + b√ - 1 . To a certain extent they suc- ceeded , but simplicity was not gained by their methods , as ...
... results of Wallis and De Moivre . They attempted to express as a line the product of two lines each represented by a symbol such as a + b√ - 1 . To a certain extent they suc- ceeded , but simplicity was not gained by their methods , as ...
Page 8
... result is obviously a numerical multiple of any one of them . Thus , if A , B , C are in one straight line , BC = xAB ; where x is a number , positive when B lies between A and C , otherwise negative : but such that its numerical value ...
... result is obviously a numerical multiple of any one of them . Thus , if A , B , C are in one straight line , BC = xAB ; where x is a number , positive when B lies between A and C , otherwise negative : but such that its numerical value ...
Page 14
... results which are sometimes useful . They may be easily verified by producing Aa to twice its length and joining the extremity with B. ( b . ) The bisectors of the sides of a triangle meet in a point , which trisects each of them ...
... results which are sometimes useful . They may be easily verified by producing Aa to twice its length and joining the extremity with B. ( b . ) The bisectors of the sides of a triangle meet in a point , which trisects each of them ...
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Common terms and phrases
a₁ axis Cartesian centre Chapter circle commutative law cone conjugate constant cöordinates coplanar curvature curve developable surface diameters differential direction drawn easily ellipsoid envelop equal evidently expression Find the equation Find the locus formula geometry given equation given lines given point given vectors gives Hamilton Hence indeterminate intersection LAOB last section length linear and vector multiply obviously origin osculating plane P₁ parabola parallel perpendicular properties prove quaternion radius rectangular represents right angles rotation S.aßy Saß scalar scalar equations second order self-conjugate sides solution sphere spherical conic 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 φρ
Popular passages
Page 153 - Find the locus of a point the ratio of whose distances from two given points is constant. Let the given points be 0 and A, the extremities of the vector a.
Page 149 - Find the equation of the locus of a point the sum of the squares of whose distances from a number of given planes is constant. 11. Substitute " lines" for "planes
Page 174 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Page 14 - The perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the vertices of the triangle. Let the -l. bisectors EE' and DD
Page 309 - Dub. Math. Journal, II. p. 62), that the forces produced by given distributions of matter, electricity, magnetism, or galvanic currents, can be represented at every point by displacements of such a solid producible by external forces. It may be useful to give his analysis, with some additions, in a quaternion form, to show the insight gained by the simplicity of the present method. Thus, if...
Page 150 - ABC, iu terms of a, /3, y. 19. Find the locus of a point equidistant from the three planes Sap = 0, Spp = 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.
Page 38 - Elementary Treatise it is accomplished by the help of the fundamental properties of the curves known as Spherical Conies, discovered only in recent times by Magnus and Chasles. Doubtless many a one has been discouraged from the study of quaternions by the abstruse nature of the fundamental principles. It is clear from the figure that the summing of versors cannot be adequately represented by a versor rotating a line...
Page 23 - Then, generally, p may be expressed as the sum of a number of terms, each of which is a...