Article 20
The Gudermanian Function.

The correspondence of sectors of the same species was discussed in Arts. 1–4. It is now convenient to treat of the correspondence that may exist between sectors of different species.

Two points $P_1,P_2$, on any hyperbola and ellipse, are said to correspond with reference to two pairs of conjugates $O_1{A}_1,O_1{B}_1$, and $O_2{A}_2,O_2{B}_2$, respectively, when

and when $y_1,y_2$ have the same sign. The sectors $A_1{O}_1{P}_1,A_2{O}_2{P}_2$ are then also said to correspond. Thus corresponding sectors of central conics of different species are of the same sign and have their primary characteristic ratios reciprocal. Hence there is a fixed functional relation between their respective measures. The elliptic sectorial measure is called the gudermanian of the corresponding hyperbolic sectorial measure, and the latter the anti-gudermanian of the former. This relation is expressed by