To ﬁnd the equation of the curve which possesses the property that the length of the tangent from the point of contact to the axis of is constant.
Let , be two consecutive tangents such that , and let ; draw perpendicular to ; then if , it is evident that diﬀers from by an inﬁnitesimal of a higher order. Let make an angle with , the axis of ; then (to the ﬁrst order of inﬁnitesimals) ; that is,
This is a convenient single-parameter form, which gives all values of , as increases from to . The value of , expressed in the same form, is found from the relation
At the point , , , , , . The Cartesian equation, obtained by eliminating , is
If be put for , and be taken as independent variable, , , ,