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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 ${O}_{1}{A}_{1},{O}_{1}{B}_{1}$ be conjugate radii of a central conic, and ${O}_{2}{A}_{2},{O}_{2}{B}_{2}$ conjugate radii of any other central conic of the same species; let ${P}_{1},{P}_{2}$ be two points on the curves; and let their coordinates referred to the respective pairs of conjugate directions be $({x}_{1},{y}_{1}),({x}_{2},{y}_{2})$; then, by analytic geometry,

$$\frac{{x}_{1}^{2}}{{a}_{1}^{2}}\pm \frac{{y}_{1}^{2}}{{b}_{1}^{2}}=1,\phantom{\rule{2em}{0ex}}\frac{{x}_{2}^{2}}{{a}_{2}^{2}}\pm \frac{{y}_{2}^{2}}{{b}_{2}^{2}}=1.$$ | (1) |

Now if the points ${P}_{1},{P}_{2}$ be so situated that

$$\frac{{x}_{1}}{{a}_{1}}=\frac{{x}_{2}}{{a}_{2}},\phantom{\rule{2em}{0ex}}\frac{{y}_{1}}{{b}_{1}}=\frac{{y}_{2}}{{b}_{2}},$$ | (2) |

the equalities referring to sign as well as magnitude, then ${P}_{1},{P}_{2}$ are called corresponding points in the two systems. If ${Q}_{1},{Q}_{2}$ be another pair of correspondents, then the sector and triangle ${P}_{1}{O}_{1}{Q}_{1}$ are said to correspond respectively with the sector and triangle ${P}_{2}{O}_{2}{Q}_{2}$. These deﬁnitions will apply also when the conies coincide, the points ${P}_{1},{P}_{2}$ 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 ${A}_{1}{O}_{1}{P}_{1}$ is the ratio

$$\frac{\text{sector}{A}_{1}{O}_{1}{P}_{1}}{\text{triangle}{A}_{1}{O}_{1}{B}_{1}}$$ |

and is to be regarded as positive or negative according as ${A}_{1}{O}_{1}{P}_{1}$ and ${A}_{1}{O}_{1}{B}_{1}$ are at the same or opposite sides of their common initial line.

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