LEMNISCATE

, the name of a curve in the form of the figure of 8. If we call A P, x; P Q, y, and the constant line A B or A C, a; the equation , or , expressing a line of the 4th degree, will denote a lemniscate, having a double point in the point A. There may be other lemniscates, as the ellipse of Cassini, &c; but that above defined is the simplest of them. |

It easily appears that this curve is quadrable. For since , therefore the fluxion of the curve or ; the fluent of which is for the general area of the curve; which, when x is = a, becomes barely .