An Elementary Treatise on Quaternions, |
From inside the book
Results 1-5 of 27
Page xii
... involves four distinct numbers - and is thus a QUATERNION , § 47 . A quaternion regarded as the operator which turns one vector into another . It is thus decomposable into two factors , whose order is indifferent , the stretching factor ...
... involves four distinct numbers - and is thus a QUATERNION , § 47 . A quaternion regarded as the operator which turns one vector into another . It is thus decomposable into two factors , whose order is indifferent , the stretching factor ...
Page xvii
... involves a scalar equation of degree m1 , § 184 . Solution of the equation q2 = qa + b , § 185 . EXAMPLES TO CHAPTER V. • 128-130 CHAPTER VI . - GEOMETRY OF THE STRAIGHT LINE AND PLANE . EXAMPLES TO CHAPTER VI . 131-147 148-150 CHAPTER ...
... involves a scalar equation of degree m1 , § 184 . Solution of the equation q2 = qa + b , § 185 . EXAMPLES TO CHAPTER V. • 128-130 CHAPTER VI . - GEOMETRY OF THE STRAIGHT LINE AND PLANE . EXAMPLES TO CHAPTER VI . 131-147 148-150 CHAPTER ...
Page 10
... involves the following three equations among numbers , x = έ , y = n , 2 = 8 . Suppose i to be drawn eastwards , j northwards , and k upwards , this is equivalent merely to saying that if two points coincide , they are equally to the ...
... involves the following three equations among numbers , x = έ , y = n , 2 = 8 . Suppose i to be drawn eastwards , j northwards , and k upwards , this is equivalent merely to saying that if two points coincide , they are equally to the ...
Page 11
... involve the indefinite quan- tity , they are virtually equivalent to but two , as in ordinary Geometry of Three Dimensions . 29. A good illustration of this remark is furnished by the fact that the equation P = ya + xß , which contains ...
... involve the indefinite quan- tity , they are virtually equivalent to but two , as in ordinary Geometry of Three Dimensions . 29. A good illustration of this remark is furnished by the fact that the equation P = ya + xß , which contains ...
Page 37
... involve only an additional numerical factor which has reference to the length merely , and not the direction , of a vector ( § 48 ) . 57. Seeing thus that the commutative law does not in general hold in the multiplication of quaternions ...
... involve only an additional numerical factor which has reference to the length merely , and not the direction , of a vector ( § 48 ) . 57. Seeing thus that the commutative law does not in general hold in the multiplication of quaternions ...
Other editions - View all
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...