An elementary treatise on quaternions |
From inside the book
Results 1-5 of 28
Page 17
... . e . is the diameter OB drawn from the origin , and a is OA the tangent at the origin . In the figure QPat , OQ Bt2 = 2 D The secant joining the points where t has the values SECT . 31. ] VECTORS , AND THEIR COMPOSITION . 17.
... . e . is the diameter OB drawn from the origin , and a is OA the tangent at the origin . In the figure QPat , OQ Bt2 = 2 D The secant joining the points where t has the values SECT . 31. ] VECTORS , AND THEIR COMPOSITION . 17.
Page 18
... diameter is - the abscissa of the point of contact . : Otherwise the tangent is parallel to the vector a + ẞt or Bt Bt2 at + ẞt or at + hence TO = - OQ . 2 + or OQ + OP . But TP = TO + OP , 2 ( 9. ) Since the equation of any tangent to ...
... diameter is - the abscissa of the point of contact . : Otherwise the tangent is parallel to the vector a + ẞt or Bt Bt2 at + ẞt or at + hence TO = - OQ . 2 + or OQ + OP . But TP = TO + OP , 2 ( 9. ) Since the equation of any tangent to ...
Page 19
... diameter produced , the chords of contact are parallel to the tangent at the vertex of the diameter . This is also proved by a former result , for we must have OT for each tangent equal to QO . ( i . ) The equation of the chord of ...
... diameter produced , the chords of contact are parallel to the tangent at the vertex of the diameter . This is also proved by a former result , for we must have OT for each tangent equal to QO . ( i . ) The equation of the chord of ...
Page 20
... diameter on which the point lies , and its inter- section with that diameter is as far beyond the vertex as the pole is within , and vice versa . ( j ) As another example let us prove the following theorem . If a triangle be inscribed ...
... diameter on which the point lies , and its inter- section with that diameter is as far beyond the vertex as the pole is within , and vice versa . ( j ) As another example let us prove the following theorem . If a triangle be inscribed ...
Page 21
... diameters . Again , p = at + B or p = a tan x + ß cotx t evidently represents a hyperbola referred to its asymptotes . But , so far as we have yet gone , as we are not prepared to determine the lengths or inclinations of vectors , we ...
... diameters . Again , p = at + B or p = a tan x + ß cotx t evidently represents a hyperbola referred to its asymptotes . But , so far as we have yet gone , as we are not prepared to determine the lengths or inclinations of vectors , we ...
Other editions - View all
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.