An elementary treatise on quaternions |
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Page 5
... prove , it can be established , ab initio , in various ways , without even an allusion to Cartesian Geometry . As this work is written for students acquainted with at least the elements of the Cartesian method , we keep to the first ...
... prove , it can be established , ab initio , in various ways , without even an allusion to Cartesian Geometry . As this work is written for students acquainted with at least the elements of the Cartesian method , we keep to the first ...
Page 10
... prove this for the sign + , it will be equally proved for — , because before a vector ( § 20 ) merely indicates that it is to be reversed before being considered positive . - Let A , B , C , D be , in order , the corners of a ...
... prove this for the sign + , it will be equally proved for — , because before a vector ( § 20 ) merely indicates that it is to be reversed before being considered positive . - Let A , B , C , D be , in order , the corners of a ...
Page 13
... proving , in a very simple manner , numerous classes of propositions in plane and solid geometry . A very few examples , however , must suffice at this stage ; since we have hardly , as yet , crossed the threshold of the subject , and ...
... proving , in a very simple manner , numerous classes of propositions in plane and solid geometry . A very few examples , however , must suffice at this stage ; since we have hardly , as yet , crossed the threshold of the subject , and ...
Page 19
... proved by a former result , for we must have OT for each tangent equal to QO . ( i . ) The equation of the chord of contact , for the point whose vector is is thus p = pa + qß , Bt , 2 2 p = at1 ++ ya + pß ) . Suppose this to pass ...
... proved by a former result , for we must have OT for each tangent equal to QO . ( i . ) The equation of the chord of contact , for the point whose vector is is thus p = pa + qß , Bt , 2 2 p = at1 ++ ya + pß ) . Suppose this to pass ...
Page 20
... prove the following theorem . If a triangle be inscribed in a parabola , the three points in which the sides are met by tangents at the angles lie in a straight line . Since O is any point of the curve , we may take it as one corner of ...
... prove the following theorem . If a triangle be inscribed in a parabola , the three points in which the sides are met by tangents at the angles lie in a straight line . Since O is any point of the curve , we may take it as one corner of ...
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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.