Article 1
Correspondence of Points on Conics.

To prepare the way for a general treatment of the hyperbolic functions a preliminary discussion is given on the relations, between hyperbolic sectors. The method adopted is such as to apply at the same time to sectors of the ellipse, including the circle; and the analogy of the hyperbolic and circular functions will be obvious at every step, since the same set of equations can be read in connection with either the hyperbola or the ellipse.1 It is convenient to begin with the theory of correspondence of points on two central conics of like species, i.e. either both ellipses or both hyperbolas.


To obtain a definition of corresponding points, let O1A1,O1B1 be conjugate radii of a central conic, and O2A2,O2B2 conjugate radii of any other central conic of the same species; let P1,P2 be two points on the curves; and let their coordinates referred to the respective pairs of conjugate directions be (x1,y1),(x2,y2); then, by analytic geometry,

x12 a12 ±y12 b12 = 1,x22 a22 ±y22 b22 = 1. (1)

Now if the points P1,P2 be so situated that

x1 a1 = x2 a2 ,y1 b1 = y2 b2 , (2)

the equalities referring to sign as well as magnitude, then P1,P2 are called corresponding points in the two systems. If Q1,Q2 be another pair of correspondents, then the sector and triangle P1O1Q1 are said to correspond respectively with the sector and triangle P2O2Q2. These definitions will apply also when the conies coincide, the points P1,P2 being then referred to any two pairs of conjugate diameters of the same conic.

In discussing the relations between corresponding areas it is convenient to adopt the following use of the word “measure”: The measure of any area connected with a given central conic is the ratio which it bears to the constant area of the triangle formed by two conjugate diameters of the same conic.

For example, the measure of the sector A1O1P1 is the ratio

 sector A1O1P1  triangle A1O1B1

and is to be regarded as positive or negative according as A1O1P1 and A1O1B1 are at the same or opposite sides of their common initial line.