412 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Xi = sinh (j), Xi = cosh </>, 



with Xi'^—Xi^ = — 1 as the equation of the pseudo-circle. 



If now in general r be the radius of any pseudo-circle, the foregoing 

 results may readily be generalized, and we obtain the following pair 

 of equations. 



Xi = r cosh 6, 

 Xi = r sinh 4>, 



Xi = r sinh 6, Xi 



Xi = r cosh 4>, Xi 



Xi tanh d; 

 Xi tanh (f). 



(1) 



In the first case r is a (7)-vector and is a (77)-angle; in the second, 

 r is a (5)-vector and </> is a (56)-angle. We thus have equations which 

 express the relations between the hypotenuse and the sides of any 

 right triangle in terms of one angle. The inclination of the vector r 

 to the axes kj or k^ in the respective cases is the angle 



d = tanh-i"^"^ 



or 



Xi 



4) = tanh '•'— ; 

 3:4 



and the slope of r relative to the axes is the hyperbolic tangent of 

 the angle, not the trigonometric tangent. 



17. Consider next any curve of class (5). Let 



= r ^dxi^ 



dxi^ 



denote scalar arc along the curve, and let r be the radius vector from a 

 fixed origin to any point of the curve. Then the derivative 



dr 

 ds 



w 



dxi dxi 



ki —J — h kj-^ 

 d^ ds 



(2) 



is a unit vector tangent to the curve. If this vector makes the angle 

 (}) with the axis k4, so that the slope of the curve is 



V = tanh 



the components of the vector are 

 dx\ 



dxi 



— » 



dXi 



ds 



= sinh0 



vr^ 



dxi 

 ds 



= cosh 4> = 



Vl— Tj2 



(3) 



(4) 



and 



w 



vr 



(I'k, + k4). 



(5) 



