Patterns for Turning; Comprising Elliptical and Other Figures Cut on the Lathe Without the Use of Any Ornamental Chuck

This historic book may have numerous typos, missing text, images, or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1872. Not illustrated. Excerpt: ... It may perhaps be remarked, that if e be taken very large indeed with respect to b, the curve becomes, in theory, entirely convex, with convexity alternately increasing and diminishing: on the scale on which we work, on the lathe, this effect will never become apparent. If 2 6 = the eccentricity for the first cut, the curve exactly reaches the centre, giving rise to loops in number 5, 4, &c. If 2 b be greater than the eccentricity for the first cut, the curve passes beyond the centre, giving rise to double the number of loops. But if the original number of loops was odd, the loops produced by the branches of the curve on opposite sides of the centre lie on each other, and if b equals the eccentricity for the first cut, they are coincident. 15. The numbers given in the Tables in any column b = 100, 80, &c., are the same read upwards or downwards; their sum equals 2 b. The patterns are produced by making cuts on the curve, or on parts of the curve mentioned in the last article; and by repeating such cuts on the same curve, or part of a curve turned round through an angle. 16. The mathematician wül readily deduce all the properties of the curve on which the cuts lie, from the following considerations -- Let e and в be its polar co-ordinates, then e = a + b sin. m 0 where a and b are arbitrary constants, m is constant for the table of m-looped figures. Giving to sin. m в its maxima and minima values, we have ex = a + b ea = a -- b whence = ey -- 2 b or e3 > = or < 0 as ex > = or < 2 b The numbers given in each of the columns b = 100, 80, &c, in the Tables of Differences Of Eccentricity are the differences of sin. m в, where в is the angle corresponding to two consecutive numbers on the division-plate, multiplied by t...