I 4 CONCRETE REPRESENTATIONS OF 



Let PY=PX+ where e is small. 

 Then 



and (PQ)=K\og (PQ, 



Let K approach infinity and e approach zero in such a way 

 that Kf. approaches a finite limit X. 



Then 



Now to fix X we must choose some point E so that (PE)=l, 



PW 



the unit of length. Then 1=X . =^ == 



JrJi. . Jf/Ji. 



^j iT>n\ PX . EX PQ XE XQ i 



(PQ)= PE PX:QX=PE^PQ =( 



If we take P as origin =0, 



Ul oo 1 



which agrees with the ordinary expression since '=1. 



0*1 



It will be noticed that this case differs in one marked 

 respect from the case of elliptic geometry. In that system 

 there is a natural unit of length, which may be taken as the 

 length of the complete straight line the period, in fact, of 

 linear measurement ; just as in ordinary angular measurement 

 there is a natural unit of angle, the complete revolution. In 

 Euclidean geometry, however, the unit of length has to be 

 chosen conventionally, the natural unit having become 

 infinite. The same thing appears at first sight to occur in 

 the hyperbolic case, since the period is there imaginary, but, 

 K being imaginary, iK is real, and this forms a natural linear 

 standard. (Of. 27 (3).) 



10. It still remains for us to consider the cases in which 

 the absolute degenerates as an envelope to two coincident 

 points and as a locus to two straight lines which may be real, 

 coincident or imaginary. In these cases k is seen to be infinite, 



