## An Elementary Treatise on Quaternions |

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algebra axes axis Calculus Cartesian Cartesian coordinates centre of inertia Chapter circle coefficients commutative law condition cone conjugate constant coordinates coplanar course curve denotes differential direction drawn easily element ellipse ellipsoid equal equivalent evidently expression factor formula geometrical given equation given quaternions given vectors gives Hamilton Hence hyperbola indeterminate integral intersection length linear and vector linear function multiplied non-coplanar notation nullitat obtain obviously once operator origin osculating plane P. G. Tait parallel parallelogram position properties prove radius rectangular represents right angles rotation scalar scalar equations second degree self-conjugate shew shewn sides solution space sphere spherical excess straight line student suppose surface tangent plane Taylor's Theorem tensor theorem transformation triangle unit-vectors vector function versor whence write written

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Page vi - Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector} Analysis; a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann.

Page 180 - 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 70 - Law: cos a = cos b cos c + sin b sin c cos A cos b = cos c cos a + sin c sin a cos B cos c = cos a cos b + sin a sin b cos C cos A = -cos B...

Page vi - To gild refined gold, to paint the lily, To throw a perfume on the violet, To smooth the ice, or add another hue Unto the rainbow, or with taper-light To seek the beauteous eye of heaven to garnish, Is wasteful, and ridiculous excess.

Page 29 - Q' parallels be drawn to CX meeting С ¥ in E, F, G; CE, CF, CG are in continued proportion. = CV+VQ , and CE.CG = CF'; because Z'-Y'=l (Ex. 2). Ex. 5. If a chord of. a hyperbola be one diagonal of a parallelogram whose sides are parallel to the asymptotes, the other diagonal passes through the centre. Let the chord be PQ ; p, p the vectors to P and Q ; then Now when one diagonal of a parallelogram is ma + nß, the other will be ma — nß.

Page 362 - The upper or lower sign is to be taken according as the surface is concave or convex towards the incident light.

Page 1 - ... by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics. 3. Wallis, towards the end of the seventeenth century, proposed to represent the impossible roots of a quadratic equation by going out of the line on which, if real, they would have been laid off. This construction is equivalent to the consideration of J — 1 as a directed unit-line perpendicular...

Page viii - Could anything be simpler or more satisfactory? Don't you feel, as well as think, that we are on the right track, and shall be thanked hereafter? Never mind when.

Page 47 - Hence it appears that i, j, k may be substituted for i, j, k; in other words, a unit-vector when employed as a factor may be considered as a quadrantal versor whose plane is perpendicular to the vector. Of course, It follows that every vector can be treated as the product of a number and a quadrantal versor. This is one of the main elements of the singular simplicity of the quaternion calculus.

Page 13 - 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