28 CONCRETE REPRESENTATIONS OF 



therefore r '=^*> 



which agrees with the former equation if c z k and cc'p, so 



that c'*=P*=k', say. 

 /c 



Hence as the representation by circles corresponds to 

 stereographic projection, the representation by straight lines 

 corresponds to central projection. 



The transformation from the sphere to the plane is in this 

 case given by the equations 



x 



where 

 Then* 



24. To determine the distance and angle functions in this 

 representation we have first the relation between the angles 

 from 22, 



tan 0'= - tan A . J^~ k ^ . - = - tan A . r *~ = - tan <6 . J -- 

 2p(r*+k) r r z +k v r'*+k' 



1 It may be noticed that the line-element can be expressed in terms of x', y" alone 

 Thus expressing z', dz' in terms of x', y' by means of the equation x'* + y' 2 + z' 3 = S?, 

 we have 



, _ Bf(dx^ + dy' 1 ) - (y'dx' - x'd^ 

 B'-x"-y' 



Here x', y', - are the so-called Weierstrass' coordinates. Let the position of a point P 



on the sphere be fixed by its distances , ij from two fixed great circles intersecting at 

 right angles at Q, and let QP=p, all the distances being measured on the sphere along 

 arcs of great circles. Then 



, , 



On the pseudosphere the circular functions become hyperbolic functions. (See Killing, 

 Die nichteulclidiachen Raumformen, Leipzig, 1885, p. 17.) 



