On quaternions, or on a new system of imaginaries in algebra by Hamilton W.R.

By Hamilton W.R.

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1 UV . νκ. ) In this expression for a vector touching a line of curvature, or parallel to such a tangent, the two terms connected by the sign ± are easily seen to denote (on the principles of the present calculus) two equally long vectors, in the directions respectively of the projections of the two cyclic normals ι and κ on a plane perpendicular to ν; that is, on the tangent plane to the ellipsoid at the proposed point, or on any plane parallel thereto. If then we draw two straight lines through the point of contact, bisecting the acute and obtuse angles which will in general be formed at that point by the projections on the tangent plane of two indefinite lines drawn through the same point in the directions of the two cyclic normals, or in directions perpendicular to the two planes of circular section of the surface, the two rectangular bisectors of angles, so obtained, will be the tangents to the two lines of curvature: which very simple construction agrees perfectly with known geometrical results, as will be more clearly seen, when it is slightly transformed as follows.

Has been deduced and geometrically interpreted as above. 55. Another mode of investigating generally the directions of those tangential vectors τ which satisfy the system of the two conditions in art. 51, may be derived from observing that * See the Numbers of the Philosophical Magazine for June, September, and October 1847; or the Proceedings of the Royal Irish Academy for July 1846. † Another very simple construction, derived from the same quaternion equation, and serving to generate, by a moving sphere, a system of two reciprocal ellipsoids, will be given in an early Number of this Magazine.

0. ) We might also have observed that by the same article 20 (Phil. , July 1846), ιτ κ + κτ ι and therefore ι dρ κ+κ dρ ι is a vector form, and that by article 26 (Phil. ) of the lines of curvature on an ellipsoid may be thus written, S . ) or, substituting for the linear element dρ the tangential vector τ , S . ) or finally, by the principles of the same 20th article, ντ ιτ κ − κτ ιτ ν = 0. ) 48. Under this last form, it was one of a few equations selected in September 1846, for the purpose of being exhibited to the Mathematical Section of the British Association at Southampton; although it happened* that the paper containing those equations did not reach its destination in time to be so exhibited.

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