An Elementary Treatise on QuaternionsAn Elementary Treatise on Quaternions by Peter Guthrie Tait, first published in 1890, is a rare manuscript, the original residing in one of the great libraries of the world. This book is a reproduction of that original, which has been scanned and cleaned by state-of-the-art publishing tools for better readability and enhanced appreciation. Restoration Editors' mission is to bring long out of print manuscripts back to life. Some smudges, annotations or unclear text may still exist, due to permanent damage to the original work. We believe the literary significance of the text justifies offering this reproduction, allowing a new generation to appreciate it. |
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Page xiii
... right angles to each other . Calling them i , j , k , we have ¿ 2 = j2 = k2 = − 1 , ij = —ji = k , jk = -kj = i , ijk - 1 , §§ 64-71 . ki = ―ik = j , A unit - vector , when employed as a factor , may be considered as a qua- drantal ...
... right angles to each other . Calling them i , j , k , we have ¿ 2 = j2 = k2 = − 1 , ij = —ji = k , jk = -kj = i , ijk - 1 , §§ 64-71 . ki = ―ik = j , A unit - vector , when employed as a factor , may be considered as a qua- drantal ...
Page 34
... angle with each other . Thus in the annexed figure B O , B1 OB = 0,41 OA Գ if , and only if , O , B1 OB ( 1 ) ... angles is understood to in- clude similarity in direction . Thus the ro- tation about an upward axis is negative ( or right ...
... angle with each other . Thus in the annexed figure B O , B1 OB = 0,41 OA Գ if , and only if , O , B1 OB ( 1 ) ... angles is understood to in- clude similarity in direction . Thus the ro- tation about an upward axis is negative ( or right ...
Page 46
... right angles , so that the figure represents a hemisphere divided into quadrantal triangles . WY j Z j k E Thus , to show that ijk , we have , O being the centre of the sphere , N , E , S , W the north , east , south , and west , and Z ...
... right angles , so that the figure represents a hemisphere divided into quadrantal triangles . WY j Z j k E Thus , to show that ijk , we have , O being the centre of the sphere , N , E , S , W the north , east , south , and west , and Z ...
Page 47
... right angles to each other , and coinciding with the axes of rotation But if we collate and compare the equations of these versors . just proved we have ( ij = k , Yi iJ = K , Sji = - k , j I = − K , — ... ( 11 ) ( 1 ) ( 12 ) ( 10 ) ...
... right angles to each other , and coinciding with the axes of rotation But if we collate and compare the equations of these versors . just proved we have ( ij = k , Yi iJ = K , Sji = - k , j I = − K , — ... ( 11 ) ( 1 ) ( 12 ) ( 10 ) ...
Page 48
... right angles . Hence it is natural to define im as a versor which turns any vector perpendicular to i through m right angles in the positive direction of rotation about i as an axis . Here m may have any real value whatever , for it is ...
... right angles . Hence it is natural to define im as a versor which turns any vector perpendicular to i through m right angles in the positive direction of rotation about i as an axis . Here m may have any real value whatever , for it is ...
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Common terms and phrases
a₁ axis Cartesian Chapter circle commutative law cone conjugate constant cöordinates coplanar curvature curve developable surface diameters differential direction drawn easily ellipsoid envelop equal equivalent evidently expression Find the equation Find the locus formula geometry given equation given lines given point given vectors gives Hamilton Hence hyperbola indeterminate intersection last section length linear and vector multiply normal origin osculating plane P₁ parabola parallel perpendicular properties prove quaternion radius rectangular represents result right angles rotation S.aßy Sapa Saß scalar scalar equations second order self-conjugate sides solution sphere spherical conic ẞ² straight line student surface surface of revolution t₁ 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 174 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Page 72 - Law: cos a = cos b cos c + sin b sin c cos A cos b = cos c cos a + sin c sin a cos B cos c = cos a cos b + sin a sin b cos C cos A = -cos B...
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. 28. Show that the locus of the pole of the plane Sap = 1, with respect to the surface Sp<f' P = l, is a sphere, if a be subject to the condition Sa<p-'a = 0.
Page 149 - Find the locus of a point the sum of the squares of whose distances from two given points is constant.
Page 217 - To find the locus of the foot of the perpendicular drawn from the origin to a tangent plane to any surface.