An elementary treatise on quaternions |
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Page xiii
... Law of Multiplication p.qr = pq.r , §§ 57-60 . [ Digression on Spherical Conics , § 59 * . ] Quaternion addition and subtraction are commutative , § 61 . Quaternion multiplication and division are distributive , § 62 . Composition of ...
... Law of Multiplication p.qr = pq.r , §§ 57-60 . [ Digression on Spherical Conics , § 59 * . ] Quaternion addition and subtraction are commutative , § 61 . Quaternion multiplication and division are distributive , § 62 . Composition of ...
Page 11
... commutative law is true for the addition of any two vectors , and is therefore generally true . Again , whatever four points are represented by A , B , C , D , we have AD = AB + BD = AC + CD , or AB + BC + CD = AB + ( BC + CD ) = ( AB + ...
... commutative law is true for the addition of any two vectors , and is therefore generally true . Again , whatever four points are represented by A , B , C , D , we have AD = AB + BD = AC + CD , or AB + BC + CD = AB + ( BC + CD ) = ( AB + ...
Page 33
... signify that a is multiplied by q , not q multiplied by a . [ This remark is of extreme importance in quaternions , for , F as we shall soon see , the Commutative Law does SECT . 49. ] PRODUCTS AND QUOTIENTS OF VECTORS . 33.
... signify that a is multiplied by q , not q multiplied by a . [ This remark is of extreme importance in quaternions , for , F as we shall soon see , the Commutative Law does SECT . 49. ] PRODUCTS AND QUOTIENTS OF VECTORS . 33.
Page 34
Peter Guthrie Tait. as we shall soon see , the Commutative Law does not generally apply to the factors of a product . ] We have also , by § 47 , q = Tq Uq = Uq Tq , where , as before , Tq depends merely on the relative lengths of a and B ...
Peter Guthrie Tait. as we shall soon see , the Commutative Law does not generally apply to the factors of a product . ] We have also , by § 47 , q = Tq Uq = Uq Tq , where , as before , Tq depends merely on the relative lengths of a and B ...
Page 37
... commutative law does not in general hold in the multiplication of quaternions , let us enquire whether the Associative Law holds . That is , if p , q , r be three quaternions , have we p.qr = pq.r ? This is , of course , obviously true ...
... commutative law does not in general hold in the multiplication of quaternions , let us enquire whether the Associative Law holds . That is , if p , q , r be three quaternions , have we p.qr = pq.r ? This is , of course , obviously true ...
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Common terms and phrases
a₁ arcs axes axis B₁ 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 extremity 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 m₁ m₂ multiply normal obviously origin osculating plane P₁ parabola parallel 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 217 - Differentiation of the equations gives us 3p + q+l equations, linear and homogeneous in the 3m + n differentials of the scalar parameters, so that by the elimination of these we have one final scalar equation in the first case, two in the second ; and thus in each case we have just equations enough to eliminate all the arbitrary parameters.
Page 14 - The bisectors of the sides of a triangle meet in a point, which trisects each of them.
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 50 - It is curious to compare the properties of these quaternion symbols with those of the Elective Symbols of Logic, as given in BOOLE'S wonderful treatise on the Laws of Thought ; and to think that the same grand science of mathematical analysis, by processes remarkably similar to each other, reveals to us truths in the science of position far beyond the powers of the geometer, and truths of deductive reasoning to which unaided thought could never have led the logician.