Article 7
Functional Relations for Hyperbola.

The functional relations between a sectorial measure and its characteristic ratios in the case of the hyperbola may be written in the form

and these express that the ratio of the two lines on the left is a certain definite function of the ratio of the two areas on the right. These functions are called by analogy the hyperbolic cosine and the hyperbolic sine. Thus, writing $u$ for $\frac{S_1}{K_1}$ the two equations

serve to define the hyperbolic cosine and sine of a given sectorial measure $u$; and the hyperbolic tangent, cotangent, secant, and cosecant are then defined as follows:

The names of these functions may be read “h-cosine,” ”h-sine,” “h-tangent,” etc., or “hyper-cosine,” etc.