Complement and Imaginary Geometry. 

 hyperbolic functions become 



cot G(m) = sinh u i 



1/ sin G(u) = co-h u, 



cos G(«) = tanli u, 



1- sinG(u) 



601 



:ang_!G(*)]= — 

 cotiG(w)=^ tt . 



s G(m) 



tanh 



2> 



The glide rmannian complement is not a recondite function 

 useful only on rare occasions, but one of the simplest and 

 most useful of all functions. For instance, in an ellipse, 

 let a mean pro) ortional between the axes be selected as the 

 unit of length, and about the centre of the ellipse let a unit 

 circle be described as in fig. 1. The area of this circle and 



Area o<?P==w/2; oa=coshu; aP = sinhw. 



of the ellipse will be equal, and for that reason it has been 

 suggested that the diameters passing through the inter- 

 sections of the two curves should be called isocyclic dia^ 

 meters*. The acute angle between these diameters is G(t/), 

 the axes are e u and e~ u , or cot-J-G(w) and tanjG(w) ; the 

 tangent of the angle which either isocyclic diameter makes 

 with the major axis is also e~ u , and the square of the focal 

 distance is 2 cot G(2w) = 2 sinh (2u). In short, the guder- 

 mannian complement specifies the ellipse more succinctly 

 than does either the eccentricity or the ellipticity, and ought 

 to be introduced into the elementary geometry of that 

 curve. 



If the ellipse is regarded as derived by finite strain from 



* Smithsonian Math. Tables, p. xxxi. 



