Elements of Quaternions |
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Common terms and phrases
Algebra arcs axes axis chord circle collinear commutative complanar conjugate cos² cosc curve cyclic permutation diagonals diameter direction of rotation equal equation esin expression factor find the locus formula given lines given point Hence initial point length line joining mean point middle point multiplication notation Operating opposite origin parabola parallel parallelogram quadrilateral quaternion quotient reciprocal represent rhombus right angles right line Saß Saßy scalar second member sides sinb sine sphere squares ß² ẞa straight line Substituting subtraction symbol tangent TaTẞ tensors Tq)² triangle unit vectors Vaß vector areas vector perpendicular versor vertex whence zero α α α γ αβ βα тв ᎢᏰ
Popular passages
Page 191 - The locus of the foot of the perpendicular from the focus on a moving tangent is the circle on the major axis as diameter. 3. The locus of the point of intersection of perpendicular tangents is a circle with radius Va>
Page 9 - If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another! the remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have the two sides BA, AC proportional to the two CD, DE, viz.
Page 234 - P : a*. (5.) Prove that if the major axis of an ellipse is equal to twice the minor axis, a straight line, equal to half the major axis, and which moves with one end on the curve and the other on the minor axis, is bisected by the major axis. (6.) A line of constant length moves with its extremities on two straight lines at right angles to each other : show that the locus of a fixed point on the moving line is an ellipse with the segments of the line for semi-axes. (7.) Adapt the last example to...
Page 90 - Show that the sum of the squares of the diagonals of any quadrilateral is twice the sum of the squares of the lines joining the middle points of the opposite sides.
Page 234 - Find the locus of a point, the square of whose distance from a given point is proportional to its distance from a given line.
Page 31 - The bisector of an angle of a triangle divides the opposite side into segments proportional to the adjacent sides. 144. Theorem. The bisector of an exterior angle of a triangle divides the opposite side produced into segments proportional to the other two sides.
Page 232 - ... squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of opposite sides. 150. If a circle be described round the point of intersection of the diameters of a parallelogram as a centre, shew that the sum of the squares on the straight lines drawn from any point in its circumference to the four angular points of the parallelogram is constant. 151. The squares on the sides of a quadrilateral are together greater than the squares...
Page 91 - In any quadrilateral the sum of the squares on the four sides exceeds the sum of the squares on the diagonals by four times the square on the straight line joining the mid-points of the diagonals. (Let E, F be the mid-points of AC, BD ; apply Apollonius' theorem to A
Page iii - The chief aim has been to meet the wants of beginners in the class-room. The Elements and Lectures of Sir WR Hamilton are mines of wealth, and may be said to contain the suggestion of all that will be done in the way of Quaternion research and application : for this reason, as also on account of their diffuseness of style, they are not suitable for the purposes of elementary instruction. The same may be...
Page 232 - ... a. Find the distance of the particle from a vertical plane passing through the axis. Also find the greatest value of a for a given helix in order that there may be a position of equilibrium of the particle. 6. A quadrilateral figure possesses the following property; any point being taken and four triangles formed by joining this point with the angular points of the figure, the centres of gravity of these triangles lie in the circumference of a circle ; prove that the diagonals of the quadrilateral...