An elementary treatise on quaternions |
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Page 13
... intersection of its diagonals . Then which gives AO + OB = AB = DC = DO + OC , AO - OC = = DO - OB . The two vectors here equated are parallel to the diagonals re- spectively . Such an equation is , of course , absurd unless ( 1 ) The ...
... intersection of its diagonals . Then which gives AO + OB = AB = DC = DO + OC , AO - OC = = DO - OB . The two vectors here equated are parallel to the diagonals re- spectively . Such an equation is , of course , absurd unless ( 1 ) The ...
Page 14
... intersect , x p = ( 1 + x ) y + 1⁄2 ß = − ( 1 + y ) B — 2 · 3 . y 27 . Since y and ẞ are not parallel , this equation gives From these Hence 1 + x = - and = - x 2 У 2 ' · ( 1 + y ) . x = y = 3 . } } AO = ( y - ẞ ) = Aa . ( See Ex . ( b ) ...
... intersect , x p = ( 1 + x ) y + 1⁄2 ß = − ( 1 + y ) B — 2 · 3 . y 27 . Since y and ẞ are not parallel , this equation gives From these Hence 1 + x = - and = - x 2 У 2 ' · ( 1 + y ) . x = y = 3 . } } AO = ( y - ẞ ) = Aa . ( See Ex . ( b ) ...
Page 16
... intersection , Rp be laid off = PA , and RQ = qB ; Q and p lie on a fixed straight line , and the length of Qp is constant . ( e . ) To find the centre of inertia of any system . If OA a , OB = a ,, be the vector sides of any triangle ...
... intersection , Rp be laid off = PA , and RQ = qB ; Q and p lie on a fixed straight line , and the length of Qp is constant . ( e . ) To find the centre of inertia of any system . If OA a , OB = a ,, be the vector sides of any triangle ...
Page 28
... intersection is att , + B p = t1 + t t1 t and thus , for the ultimate intersections , where £ = 1 , P = + = ( at + 2 ) as before . COR . ( 1 ) . If tt , = 1 , P = a + B 1 ; t +7 or the intersection lies in the diagonal of the ...
... intersection is att , + B p = t1 + t t1 t and thus , for the ultimate intersections , where £ = 1 , P = + = ( at + 2 ) as before . COR . ( 1 ) . If tt , = 1 , P = a + B 1 ; t +7 or the intersection lies in the diagonal of the ...
Page 30
... intersections of AB , ab , of BC , bc , and of CA , ca , lie in a straight line . 5. Prove the converse of 4 , i . e ... intersection bisects the lines join- ing the points where the tangents meet the asymptotes : and the tangent at the ...
... intersections of AB , ab , of BC , bc , and of CA , ca , lie in a straight line . 5. Prove the converse of 4 , i . e ... intersection bisects the lines join- ing the points where the tangents meet the asymptotes : and the tangent at the ...
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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.