An elementary treatise on quaternions |
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Page 38
... spherical conic is the curve of intersection of a cone of the second degree with a sphere , the vertex of the cone being the centre of the sphere . If a cone have one series of circular sections , it has another series , and any two circles ...
... spherical conic is the curve of intersection of a cone of the second degree with a sphere , the vertex of the cone being the centre of the sphere . If a cone have one series of circular sections , it has another series , and any two circles ...
Page 39
... sphere whose centre M m R L is 0 , IL , Mm are the great circles which re- present the cyclic planes , PQR is the spherical conic which represents the cone . The point P represents the line OpP , and so with the others . The ...
... sphere whose centre M m R L is 0 , IL , Mm are the great circles which re- present the cyclic planes , PQR is the spherical conic which represents the cone . The point P represents the line OpP , and so with the others . The ...
Page 40
... spherical conic whose cyclic planes are those of AB , FE . Hence also KG passes through L , and with LM intercepts on AB an arc equal to AB . That is , it passes through N , or KG and LN are arcs of the same great circle and they are ...
... spherical conic whose cyclic planes are those of AB , FE . Hence also KG passes through L , and with LM intercepts on AB an arc equal to AB . That is , it passes through N , or KG and LN are arcs of the same great circle and they are ...
Page 41
... Spherical Conics already employed in de- monstrating the truth of the associative principle . For continuity we give ... conic of which BC and be are the cyclic arcs . And , because 63 = dò = cy , BE , F , G , when produced , meet in a point ...
... Spherical Conics already employed in de- monstrating the truth of the associative principle . For continuity we give ... conic of which BC and be are the cyclic arcs . And , because 63 = dò = cy , BE , F , G , when produced , meet in a point ...
Page 42
Peter Guthrie Tait. which is also on the spherical conic ( § 59 * ) . Let these arcs meet BC in J , L , K respectively . Then we have JH = E3 = p Uq , LH = Fò = p U ( q + r ) , KH = Gy = p Ur . Also and LJ = BD , KL = CD . And , on ...
Peter Guthrie Tait. which is also on the spherical conic ( § 59 * ) . Let these arcs meet BC in J , L , K respectively . Then we have JH = E3 = p Uq , LH = Fò = p U ( q + r ) , KH = Gy = p Ur . Also and LJ = BD , KL = CD . And , on ...
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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.