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a continuous contour is obtained in this manner, and the two spirals can be drawn by the method above described (185).

The dotted line cba pa' d' indicates another pair of spirals drawn around the first pair, having the same poles S and S and the same axes SX and SX', but measured on different scales. In the figure Sp is taken equal to S'P, and S'p equal to SP; but any variation can be made in the scales employed.

187. To find points upon a spiral in which the radiusvector varies inversely as the cube-root of the angle of revolution from a fixed axis. Let BSX (fig. 153) be the fixed axis, S the pole, CSA at right-angles to the axis. Take SX equal to 1000 on any scale, and from S as a centre, with SX as radius, describe a circle; from X as a centre, with 959 on the same scale as radius, draw an arc cutting the circle in K, which is a point on the curve; or K will be found by making the angle KSX equal to 57.3o. Draw SP, making with the axis an angle of 27°, and measure SP equal to 1285, then P is the point of contraryflexure, the curve becoming convex to the axis as it approaches it from the point P; and the axis is an asymptote (124) to the curve.

To find other points on the spiral, divide each quadrant into four equal angles, and measure the lengths from S as marked on the figure by the foregoing scale. Thus SJ bisects the angle ASX and measures 1084 on the same scale as SK measures 1000; bisect JSX by the line SG, and measure SG equal to 1366. Bisect the angle GSX by the line SL, and measure SL equal to 1721; bisect LSX by SM, and measure SM equal to 2167. The line SH. bisects the angle JS A, and is equal to 947.

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This spiral can be made into a double spiral in the same manner as described above (186), by joining on another similar curve at the point P, and using either the same or a different scale for the two spirals.

188. To draw tangents at various points of the foregoing spiral.-Let $ represent the angle between the tangent and radius-vector at any point on the curve, O the angle (in circular measure) which the radius-vector has revolved over from the fixed axis ; then it is found by analysis that

tan $=3 4.

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Froin this formula we can draw the tangent at any given point on the curve. Thus at L the tangent T L is drawn by making Ll perpendicular to the radius-vector SL, and equal to 59 on any scale; then take IT equal to 100 on the same scale. At P, we find Pp to Tp as 1414 to 1000, and T P is the tangent at P; proceeding in the same way at K, we find Kk to Tk as 3 to 1; at A, Aa is to Ta as 471 to 100; at E, Ee is to Te as 707 to 100; at B, B b is to Tb as 942 to 100; and at C, Cc is to Tc as 1414 to 100.

After passing C the tangents become very nearly perpendicular to the radius-vector, and the curvature approaches that of a circle.

Having drawn the tangents the curve can readily be described by hand through the several points; or by drawing normals perpendicular to the tangents at those points, arcs of circles can be described from point to point having their centres at the intersections of consecutive normals.

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THE EQUIANGULAR OR LOGARITHMIC SPIRAL.

189. To find points on the equiangular spiral, the depth and aris of which are given.—Let DSC (fig. 154) be the given axis, S the pole, A S B the given depth ; the ratio of A S to BS being also given. Find SC a mean-proportional

Fig. 154.

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(17) between AS and BS, then C is a point where the spiral cuts the axis. Join A C, and draw B D parallel to AC cutting the axis in D, then D is also a point on the curve. Join BC, and draw D E parallel to BC, and E is

point where the spiral cuts SA. Draw EG parallel to

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