An Elementary Treatise on Quaternions, |
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Page 12
... indeterminate number ; and a plane , if they be linear expressions containing two indeterminate numbers . Later ( § 31 ( 7 ) ) , this theorem will be much extended . 30. The equation of the line joining any two points A and B , where OA ...
... indeterminate number ; and a plane , if they be linear expressions containing two indeterminate numbers . Later ( § 31 ( 7 ) ) , this theorem will be much extended . 30. The equation of the line joining any two points A and B , where OA ...
Page 22
... indeterminate . In such a case the equation is sometimes written p = $ ( t ) . But , if P1 , P2 , & c . be functions of two indeterminates , the locus of the extremity of p is a surface ; whose equation is sometimes written p = $ ( t ...
... indeterminate . In such a case the equation is sometimes written p = $ ( t ) . But , if P1 , P2 , & c . be functions of two indeterminates , the locus of the extremity of p is a surface ; whose equation is sometimes written p = $ ( t ...
Page 23
... indeterminate ; or , as in § 31 ( 1 ) , if P be a point on the curve , OP = p = $ ( t ) . And , similarly , if Q be any other point on the curve , OQ = p1 = $ ( t1 ) = $ ( t + dt ) , where it is any number whatever . The vector - chord ...
... indeterminate ; or , as in § 31 ( 1 ) , if P be a point on the curve , OP = p = $ ( t ) . And , similarly , if Q be any other point on the curve , OQ = p1 = $ ( t1 ) = $ ( t + dt ) , where it is any number whatever . The vector - chord ...
Page 27
... indeterminate . and From these , at once , x = Thus the equation of the envelop is p = at +1 ( at ) = } ( at + 2 ) , the hyperbola as before ; a , ß being portions of its asymptotes . 42. It may assist the student to a thorough ...
... indeterminate . and From these , at once , x = Thus the equation of the envelop is p = at +1 ( at ) = } ( at + 2 ) , the hyperbola as before ; a , ß being portions of its asymptotes . 42. It may assist the student to a thorough ...
Page 28
... indeterminate . At the point of intersection of these lines we have ( § 26 ) , t ( 1 - x ) = t1 ( 1 — y ) , ' Ꮖ y t = These give , by eliminating y , t1 t1 t ( 1 − x ) = t , ( 1 − 1 ; x ) , - t - or x = t1 + t Hence the vector of the ...
... indeterminate . At the point of intersection of these lines we have ( § 26 ) , t ( 1 - x ) = t1 ( 1 — y ) , ' Ꮖ y t = These give , by eliminating y , t1 t1 t ( 1 − x ) = t , ( 1 − 1 ; x ) , - t - or x = t1 + t Hence the vector of the ...
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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.