SPIRAL

In Geometry, the name given to a cla of curves distinguished by this general property, that they continually recede from a centre or pole, while they continue to revolve about it. Spirals receive different names from the properties by which they are characterised, or from their inventors; thus, the equable spiral, the hyperbolic spiral, the logarithmic Spiral, the spirals of Cotes, ^ y between the pole and any point of the curve is the radiant, or radius vector at that point. Let U be a point in the spiral, make the radiant P(l=«, and let 0 be the angle described by the revolving line while the travelling point advances from P to Q.; then, since from the definition the radiant u and the angle 0 are both proportional to the time, they are proportional to each other; and the polar equation of the equable spiral is u = a 6, where c is a constant. If we suppose u-=.r when the revolving liut has made a complete revolution, or when 6 — 2 77; thet a = r -7- 2 r, and the equation becomes u = 5— 0. Some of the principal properties of this spiral are the following:1.The area of the spiral P a A b Q. is equal to 6«3a _ ^3 r If u _ thjg becomes i „ T2; or the area generated while the revolving line makes one revolution, is equal to a third of a circle whose radius is r. If u = 2r, the expre ion for the area becomes g ti• (2r)Z; and the spiral area is therefore the third part of a space that is double of a circle described with a radius = 2 r. And generally the whole area generated by the radiant, from the beginning of the motion till part after of a any space number which of revolu-is the tions,is equal to the third same multiple of the circle whose radius is equal to the greatest radiant as the number of revolution is ol mnt. 2 The arc of the spiral between its origin and the radiant u U [s. 1161]