An Elementary Treatise on Quaternions, |
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Page xiv
... space indifferently in all directions , § 93 . EXAMPLES TO CHAPTER II . . 59 , 60 CHAPTER III . - INTERPRETATIONS AND TRANSFORMATIONS OF QUATERNION EXPRESSIONS . If be the angle between two vectors , a and B , we have • 61-85 B TB S cos ...
... space indifferently in all directions , § 93 . EXAMPLES TO CHAPTER II . . 59 , 60 CHAPTER III . - INTERPRETATIONS AND TRANSFORMATIONS OF QUATERNION EXPRESSIONS . If be the angle between two vectors , a and B , we have • 61-85 B TB S cos ...
Page 4
... space . 10. In the present century Argand , Warren , and others , ex- tended the results of Wallis and De Moivre ... space the form a + b√1 for the plane , he is guided by analogy to write for a directed unit - line in space the form p ...
... space . 10. In the present century Argand , Warren , and others , ex- tended the results of Wallis and De Moivre ... space the form a + b√1 for the plane , he is guided by analogy to write for a directed unit - line in space the form p ...
Page 5
... space expressible by real numbers , the remainder being imaginaries of some kind , leading in general to equations which are heterogeneous ; Hamilton makes all di- rections in space equally imaginary , or rather equally real , thereby ...
... space expressible by real numbers , the remainder being imaginaries of some kind , leading in general to equations which are heterogeneous ; Hamilton makes all di- rections in space equally imaginary , or rather equally real , thereby ...
Page 6
... space ; it is only the relative coordinates that are contemplated . Hence any expression , as AB , denoting a line considered with reference to direction as well as length , contains implicitly three numbers , and all lines parallel and ...
... space ; it is only the relative coordinates that are contemplated . Hence any expression , as AB , denoting a line considered with reference to direction as well as length , contains implicitly three numbers , and all lines parallel and ...
Page 11
... space . As we have seen ( § 25 ) it is equivalent to three numerical equations ; but , as these involve the indefinite quan- tity , they are virtually equivalent to but two , as in ordinary Geometry of Three Dimensions . 29. A good ...
... space . As we have seen ( § 25 ) it is equivalent to three numerical equations ; but , as these involve the indefinite quan- tity , they are virtually equivalent to but two , as in ordinary Geometry of Three Dimensions . 29. A good ...
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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...