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angle application axes axis becomes centre Chapter circle common condition cone conjugate considered constant contains course curvature curve definite determine developable surface diameters differential direction distance drawn easily effect element Eliminating ellipsoid employed envelop equal equation equivalent evidently example expression extremity fact factors fixed formula function given gives Hamilton Hence indeterminate integral interpretation intersection involves joining last section length locus means meet multiply normal obtain obviously once Operating origin parallel pass perpendicular plane positive proof properties prove quaternion radius rectangular referred represents respectively result right angles rotation satisfied scalar second order seen sides similar Similarly simple solution space sphere spherical student suppose surface tangent plane tensor transformation triangle unit-vector vector vector function versor write written
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 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...