An Elementary Treatise on Quaternions, |
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Page 18
... determine t and x , t + x = p , 12 + xt = 9 ; 2 by equating respectively the coefficients of a and B. Hence t = p ± √ p2 -2q . Thus , in general , two tangents can be drawn 18 [ CHAP . I. QUATERNIONS . Expression of a vector by one ...
... determine t and x , t + x = p , 12 + xt = 9 ; 2 by equating respectively the coefficients of a and B. Hence t = p ± √ p2 -2q . Thus , in general , two tangents can be drawn 18 [ CHAP . I. QUATERNIONS . Expression of a vector by one ...
Page 21
... determine the lengths or inclinations of vectors , we can only investigate a very small class of the properties of curves , re- presented by such equations as those above written . ( 7. ) We may now , in extension of the statement in ...
... determine the lengths or inclinations of vectors , we can only investigate a very small class of the properties of curves , re- presented by such equations as those above written . ( 7. ) We may now , in extension of the statement in ...
Page 54
... determines w . Also we have , from the same three , Ex2 + ny ′ + 8 = 0 ; which , combined with the first , gives ૐ = yz - zy η - = * xz ' xy ' — yx ' ' - and the common value of these three fractions is then easily seen to be 1 x2 + y2 ...
... determines w . Also we have , from the same three , Ex2 + ny ′ + 8 = 0 ; which , combined with the first , gives ૐ = yz - zy η - = * xz ' xy ' — yx ' ' - and the common value of these three fractions is then easily seen to be 1 x2 + y2 ...
Page 58
... determine the direction in which it must lie . It cannot be a vector parallel to them ; for by changing the sign of both factors the product is unchanged , whereas , as the whole system has been reversed , the product vector ought to ...
... determine the direction in which it must lie . It cannot be a vector parallel to them ; for by changing the sign of both factors the product is unchanged , whereas , as the whole system has been reversed , the product vector ought to ...
Page 59
... determine the rule which connects the directions in which these angles are to be measured . 2. Hence derive another proof that we have not generally pq = qp . 3. Hence show that the proof of the associative principle , § 57 , may be ...
... determine the rule which connects the directions in which these angles are to be measured . 2. Hence derive another proof that we have not generally pq = qp . 3. Hence show that the proof of the associative principle , § 57 , may be ...
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Common terms and phrases
a₁ axis 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 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 multiply obviously origin osculating plane P₁ parabola parallel perpendicular 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 174 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Page 14 - The perpendicular bisectors of the sides of a triangle meet in a point which is equidistant from the vertices of the triangle. Let the -l. bisectors EE' and DD
Page 309 - Dub. Math. Journal, II. p. 62), that the forces produced by given distributions of matter, electricity, magnetism, or galvanic currents, can be represented at every point by displacements of such a solid producible by external forces. It may be useful to give his analysis, with some additions, in a quaternion form, to show the insight gained by the simplicity of the present method. Thus, if...
Page 150 - ABC, iu terms of a, /3, y. 19. Find the locus of a point equidistant from the three planes Sap = 0, Spp = 0, Syp = 0. 20. If three mutually perpendicular vectors be drawn from a point to a plane, the sum of the reciprocals of the squares of their lengths is independent of their directions.
Page 38 - Elementary Treatise it is accomplished by the help of the fundamental properties of the curves known as Spherical Conies, discovered only in recent times by Magnus and Chasles. Doubtless many a one has been discouraged from the study of quaternions by the abstruse nature of the fundamental principles. It is clear from the figure that the summing of versors cannot be adequately represented by a versor rotating a line...
Page 23 - Then, generally, p may be expressed as the sum of a number of terms, each of which is a...