An Elementary Treatise on Quaternions, |
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Page 26
... obtain afresh the above result without any reference to time or velocity . 39. Following strictly the process of Newton's VIIth Lemma , let us describe on Pq , an arc similar to PQ ,, and so on . Then obviously , as the subdivision of ...
... obtain afresh the above result without any reference to time or velocity . 39. Following strictly the process of Newton's VIIth Lemma , let us describe on Pq , an arc similar to PQ ,, and so on . Then obviously , as the subdivision of ...
Page 27
... − x ) ) = 0 , = 0 ; and these are , when expanded , the equations we obtained in the preceding section . 43. For farther illustration we give a solution not directly E 2 SECT . 42. ] VECTORS , AND THEIR COMPOSITION . 27.
... − x ) ) = 0 , = 0 ; and these are , when expanded , the equations we obtained in the preceding section . 43. For farther illustration we give a solution not directly E 2 SECT . 42. ] VECTORS , AND THEIR COMPOSITION . 27.
Page 67
... obtaining these results is to expand the above expression , thus , § 90 ( 2 ) p = 2a - 1Saẞ - ß = 2a - 1Saẞ - a - 1 ( Saß + Vaß ) = a 1 ( Saß - Vaß ) , so that in the figure of § 77 we see that if OA = a , and OB = ß , we have OD = p ...
... obtaining these results is to expand the above expression , thus , § 90 ( 2 ) p = 2a - 1Saẞ - ß = 2a - 1Saẞ - a - 1 ( Saß + Vaß ) = a 1 ( Saß - Vaß ) , so that in the figure of § 77 we see that if OA = a , and OB = ß , we have OD = p ...
Page 71
... obtain the results as in ordinary trigonometry . From De Moivre's Theorem , thus proved , we may of course deduce the rest of Analytical Trigonometry . And as we have already deduced , as interpretations of self - evident quaternion ...
... obtain the results as in ordinary trigonometry . From De Moivre's Theorem , thus proved , we may of course deduce the rest of Analytical Trigonometry . And as we have already deduced , as interpretations of self - evident quaternion ...
Page 77
... obtained when we extend these processes of interpretation to functions of a quaternion q = w + p instead of functions of a mere vector p . A scalar equation containing such a quaternion , along with quaternion constants , gives , as in ...
... obtained when we extend these processes of interpretation to functions of a quaternion q = w + p instead of functions of a mere vector p . A scalar equation containing such a quaternion , along with quaternion constants , gives , as in ...
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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...