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radius vector with the fixed axis.—Let S (figs. 145, 146,

Fig. 145.

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147, 148) be the pole, SX the fixed axis. Through: S draw a line at right-angles to S X, and divide each quadrant into four equal angles. Measure from S along these lines the cubes of all successive integers, beginning with unity, as given in the Table (181), on any convenient scale ; sixteen points on each convolution will be thus obtained. The numbers marked on the figures themselves give the dis

Fig. 146.

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tances of the several points from the pole, and the spiral can be drawn by hand through the points thus found.

Figure 145 shows the spiral for the first half-convolution starting from the pole.

Figure 146 gives the first whole convolution of the spiral from the pole on a scale one-twentieth of that used in the former figure,

Figure 147 shows three and a quarter convolutions drawn to a scale one-twentieth of that used in figure 146.

Figure 148 gives six and a quarter convolutions from the pole, drawn to a scale one-tenth of that used in drawing figure 147. .

179.

To draw the tangents at various points of the fore

Fig. 147.

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going spiral.--Let $ be the angle which the tangent and radius-vector make with each other at any given point on the curve, and let 6 be the angle (in circular measure) which the radius-vector at that point has revolved over from the axis ; then it is found by analysis that

tan 0= 19.

The angle is acute and less than half a right-angle in the first half-convolution from the pole, and after that becomes greater than half a right-angle, increasing as the curve proceeds, and getting nearer and nearer to a rightangle.

To draw the tangent at A (fig. 145), take A a perpendicular to SA, and equal in length to 53 on any convenient scale, erect a T at right-angles to a A and equal to 100 on the same scale, then TA is the tangent at A. Proceeding in the same manner we find the tangent át B by taking Bb to Tb as 79 to 100; at C, take Cc to Tc as 105 to 100, and TC is the tangent.

To draw the tangent at D (fig. 146), take Dd to Td as 157 to 100; at E, take Ee to Te as 209 to 100; at F (fig. 147), take Ff to Tf as 262 to 100; at G, take Gg to Tg as 314 to 100; at H, take Hh to Th as 373 to 100; at I, take I i to Ti as 419 to 100; at J, take J i to Ti as 471 to 100; at K, take Kk to Tk as 524 to 100; at L, take Ll to Tlas 587 to 100; at M, take Mm. to T m as 628 to 100; and at N, take Nn to Tn as 694 to 100; then TN will be the tangent at N.

180. To apply the above spiral to form a volute of given dimensions.—Let Y Z (fig. 148) be the given height of the volute. Divide Y Z at S, so that SZ shall be to SY in the ratio of the cube of 100 to the cube of 92 (see Table of

cubes (181) ), or nearly as 100 to 78. Then draw W SX at right-angles to Y Z, and divide each quadrant into four equal angles by lines drawn through the pole S; take SX

Fig. 148.

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to SW as the cubes of 96 and 98, or nearly as 881 to 68, and measure on the other lines the lengths as given in the table of cubes, starting from SZ as the cube of 100, then SU the cube of 99, SV the cube of 98, and so on for as many convolutions as may be required.

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