SPIRAL SURFACES.

211

TWELFTH LESSON.

Spiral Surfaces.

Before explaining the properties of spiral surfaces,and the applications of them in the arts, it is necessaryto examine the curves by which such surfaces are pro-duced.

Having drawn the rectangle OHk, fig. 1, pi. 12, letus divide it into bands of an equal breadth, by meansof the parallel right lines A b, Be, C d, &c. Draw theoblique lines A a, B b, Cc, T>d, which will be parallel toeach other, for they intercept equal portions of the otherparallels, AB = ab, BC = be, CD = cd , &c.

Let us suppose that the rectangle is bent into anycylindrical form, having OH for one of its edges. Shutup the cylinder, so that ak will coincide exactly with OH.The point a will fall on the point O, b will fall on A, d onC, &c. The edges being all parallel to OA and ak, willbe represented in the rectangle OHfta, by the right linesPQ, RS, TU, &c., parallels to the sides OHa/c. Butin the rectangle, all these parallel right lines intersect theobliques A a, \\b, Cc, under the same angle, for all theseobliques are parallels. If we bend the rectangle on thecylinder, fig. 3, none of the angles formed by the obliques

r 2