A Geometrical Determination of the Canonical Quadric of by Stouffer E. B.

By Stouffer E. B.

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Take the relation sin Π(a) = sin B . cos A Let p, q, and r be the sides of the triangle AB C of our last demonstration and p , q , and r the corresponding sides of the triangle formed in the same way on a boundary-surface tangent to the plane ABC at B. 2 We can arrange the parts of a right triangle so as to apply Napier’s rules; namely, the arrangement would be CHAPTER 3. THE HYPERBOLIC GEOMETRY 45 q , r q cos A = . r q ∴ sin Π(a) = . q sin B = Now q and q are corresponding arcs on two boundary-curves which have the same set of parallel lines as axes, and their distance apart, x, is the distance from a boundary-curve of the extremity of a tangent of arbitrary length, a.

The two areas are then to each other in the ratio s rn − 1 rm − 1 :t . r−1 r−1 But s rm = s and t rn = t, so that this is the same as the ratio s−s : t−t. When l and k are incommensurable, we proceed as in other similar demonstrations. This theorem is analogous to the one which we have proved about polygons: the area is proportional to the amount of rotation in excess of four right angles in going around the figure, for the rate of rotation in going along a boundary-curve is constant. The locus of points at a given distance from a straight line is a curve which may be called an equidistant-curve.

Two points which are equidistant from one line are equidistant from its polar. The locus of points which are at a given distance from a fixed line is a surface of revolution having both this line and its polar as axes. We may call it a surface of double revolution. The parallel circles about one axis are meridian curves for the other axis. If a solid body, or, we may say, all space, move along a straight line without rotating about it, it will rotate about the conjugate line as an axis without sliding along it.

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