NON-EUCLIDEAN GEOMETRY n 



Eliminating ra we find the equation of the locus of P, 



which is a conic having double contact with the absolute at 

 A and B. 



The equation of the tangent at P^y^Zj) is 



and that of OP is 

 The pole of the line OP with respect to the absolute is (x v 

 y lt 0), and this lies on the tangent. Hence OP and the 

 tangent are conjugate with respect to the absolute and are 

 therefore at right angles. 



6. When the absolute is imaginary X, Y are conjugate 

 imaginary points, and log (PQ, X Y) is a pure imaginary. In 

 order that the distance may be real, K must then be a pure 

 imaginary, and, as in the case of angles, we see that distance 

 is a periodic function with period 2-n-Ki. By taking K=%i the 

 period becomes IT, and we make linear measurement correspond 

 with angular. This case will be seen to correspond to spherical 

 geometry, but the period (the radius of the sphere being unity) 

 is not TT but 2ir. This is exactly analogous to the case of two 

 rays, or lines with defined sense. On the sphere two antipodal 

 points define the same pencil of great circles, but with opposite 

 sense of rotation. If we leave the sense of rotation undefined, 

 then they determine exactly the same pencil, and must be 

 considered identical, or together as forming a single point ; 

 just as two rays, which make an angle TT, together form a 

 single line. On the sphere two lines (great circles) determine 

 two antipodal points or pencils of opposite rotations ; two 

 points determine two rays of opposite directions. It is 

 convenient thus to consider antipodal points as identical, or 

 we may conceive a geometry in which this is actually the case. 

 This is the geometry to which the name elliptic is generally 

 confined, the term spherical being retained for the case in 



