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abcd arbitrary vector axes axis base-point centre of mass circle commutative Compare Art condition cone confocals const constant coplanar corresponding curvature curve determined differential equation direction edges Elements equal expression extremity Find the locus fixed point forces function f given Hamilton Hence homography imaginary impulsive wrench instantaneous integral intersection invariant inverse last article latent roots linear function linear vector function locus mutually rectangular normal operator origin osculating plane pair parallel parallelepiped parameters perpendicular positive principal curvatures Prove quadratic quadric quadric surface quaternion radius ratio reciprocal reduced relation represents respect right angles rigid body rotation round ruled surface satisfied scalar function screw sphere spin-vector square surface surface integral symbols tangent plane tensor tetrahedron three mutually transformation triangle twist velocity twisted cubic unit vectors values vanishes variable vector area versor write zero
Page 114 - From a fixed point A, on the surface of a given sphere, draw any chord AD; let D...
Page 114 - If be the second point of intersection of the sphere with the secant BD drawn from any point B ; and take a radius vector AE, equal in length to BD', and in direction either coincident with, or opposite to, the chord AD : the locus of E is an ellipsoid, whose centre is A, and which passes through B. (Hamilton, Elements, p. 227.) 23. Show that the equation l1 (e...
Page 211 - The bare inspection of these forms may suffice to convince any person who is acquainted, even slightly (and I do not pretend to be well acquainted), with the modern researches in analytical physics, respecting attraction, heat, electricity, magnetism, etc., that the equations of the present article must yet become (as above hinted) extensively useful in the mathematical study of nature, when the calculus of quaternions shall come to attract a more general attention than that which it has hitherto...
Page 1 - CHAPTER VIII. IMPACT OF WATER ON VANES. 158. Definition of a vector. A right line AB, considered as having not only length, but also direction, and sense, is said to be a vector*. The initial point A is said to be the origin. It is important that the difference between sense and direction should be clearly recognised. Suppose for example, from any point A, a line AB of definite length is drawn in a northerly direction, then the direction of...
Page 2 - The sum is the segment extending from the initial point of the first to the terminal point of the second.
Page 184 - ... space." The force is directed toward the center, as usual. It may be observed that if in general the force is central, the moment of momentum is constant. For if dt (mv) = I, iv<^ (wiv) = ^ (ivmv) = r,xf = 0.
Page viii - Macfarlane, published by the International Association for the promotion of the Study of Quaternions and Allied Systems of Mathematics (Dublin, 1904), renders unnecessary nny detailed list of works on quaternions.
Page 30 - Cn2 cos c — 2 cos a . cos 6 ; from which we obtain, cos c — cos a . cos b cos c = : sin a . sin b This formula is inconvenient for logarithmic computation. /or...
Page 30 - LMN is equal to (-R2 — d2) sin A sin B sin C where R is the radius of the circle circumscribing the triangle ABC, and d the distance of its centre from P.
Page 211 - Plane ; intending then, as a ninth andfínal specimen, to giye briefly a quaternion transformation of a celebrated equation in partial differential coefficients, of the first order and second degree, which occurs in the theory of heal, and in that of the attraction of spheroid».