The class of curves resulting from the formula

are of considerable interest as occurring in various mechanical inquiries. When a straight wire, whose effective breadth and thickness are as two to one, is fixed at one end and made to vibrate, its free end describes a curve of which the general equation is

in which k is constant for the particular variety of curve. When the curve becomes a parabola, and when k = 0, it takes the form above mentioned; these phases were exhibited by me in 1832. Again, when a system of toothed wheels is deduced from a straight rack, having a curve of sines for its outline, the points of contact describe a curve of this class, as is shown in my treatise on the teeth of wheels.