An Elementary Treatise on Quaternions, |
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Page 30
... circle through an angle positive direction of rotation , without altering its length , deduce the ordinary formulæ for cos ( A + B ) , cos ( 4 - B ) , sin ( A + B ) , and sin ( A - B ) , in terms of sines and cosines of A and B. 8. If ...
... circle through an angle positive direction of rotation , without altering its length , deduce the ordinary formulæ for cos ( A + B ) , cos ( 4 - B ) , sin ( A + B ) , and sin ( A - B ) , in terms of sines and cosines of A and B. 8. If ...
Page 36
... circle , of which the two vectors are radii . This is of considerable importance in the proofs which follow . A , Thus the versor may be repre- OB 01 sented by the arc AB , which written AB . And , similarly , the versor may be OB is OA ...
... circle , of which the two vectors are radii . This is of considerable importance in the proofs which follow . A , Thus the versor may be repre- OB 01 sented by the arc AB , which written AB . And , similarly , the versor may be OB is OA ...
Page 37
... circle . But rq is easily seen to be represented by the arc AE . For whence 201 qOA OB , and rOB = rqOA OE , and ... circles , for their tensors may obviously ( § SECT . 58. ] PRODUCTS AND QUOTIENTS OF VECTORS . 37.
... circle . But rq is easily seen to be represented by the arc AE . For whence 201 qOA OB , and rOB = rqOA OE , and ... circles , for their tensors may obviously ( § SECT . 58. ] PRODUCTS AND QUOTIENTS OF VECTORS . 37.
Page 38
Peter Guthrie Tait. great circles , for their tensors may obviously ( § 48 ) be divided out from both sides , being commutative with the versors . Let AB = p , ED = CA = q , and FE = r . Join BC and produce the great circle till it meets ...
Peter Guthrie Tait. great circles , for their tensors may obviously ( § 48 ) be divided out from both sides , being commutative with the versors . Let AB = p , ED = CA = q , and FE = r . Join BC and produce the great circle till it meets ...
Page 39
... circle ( that in which its plane cuts the sphere A ) and the exterior angle at p is equal to the interior angle at Q ... circles which re- present the cyclic planes , PQR is the spherical conic which represents the cone . The point P ...
... circle ( that in which its plane cuts the sphere A ) and the exterior angle at p is equal to the interior angle at Q ... circles which re- present the cyclic planes , PQR is the spherical conic which represents the cone . The point P ...
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Common terms and phrases
a₁ axis B₁ Cartesian centre Chapter circle commutative law cone conjugate constant cöordinates coplanar curve diameters differential direction drawn easily ellipsoid envelop equal evidently expression extremity Find the equation Find the locus formula geometry given equation given lines given vectors gives Hamilton Hence hyperbola indeterminate intersection inverse function LAOB last section length linear and vector m₁ multiply obviously origin osculating plane P₁ parabola parallel perpendicular properties prove quaternion radius rectangular represented right angles rotation S.aßy Saß scalar scalar equations second order self-conjugate sides solution Spdp sphere spherical conic Spopp 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 Γλμ 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.