the Projective and the Metrical Scales. 425 

 parallels, put I — go . Then, since Tan x> =1, 



Tan2\ — T an\ ,, 



n ~TanX(Tan-2X-l)"- " ' ' " ' vO 



Similarly, for I— — oo , 



, Tan 2\-TanX , 



n_ TanX(Tan2X + l) {i) 



These two different indices correspond to the two " points 



at infinity " (or pencil centres) of the Lobatchevskyan straight 



line. 



e 2x — 1 

 Remembering that Tana?=- r, the last formulae are 



easily transformed into 

 giving the simple relation 



*m'=l (9) 



Now, consider the two points P and P' (fig. 5) ; whose 

 Staudtian indices are jdie negatives of those in (8), 



v = e 2 \ and v[=e'^. ..... (10) 



Fig. 5. 



V f MV T 



■- - ■■■ ■ O " '■'■■'■ ■ ■ ■'■ © ■' o ■ < » ■ ■■■ ■ — o — , 



o y*l y oo 



In order to find their distances from the origin, l=OP, 

 and V — OP\ substitute the indices (10) into (6), with 

 £ oo =2Zi = 2\, as before. Then the result will be 



The metrical distance OJ/ being X, let us find the length 

 P'M=\ — V. Remembering that 



Tan(\-/') = (Tan\-TanZ') : (1-TanX .Tan V), 



we shall find, after easy reductions, 



Tan(V-Z') = Tan 2 \. 



And, in quite the same way, for MP — l—X, 



Tan(2-\) = Tan 2 \, 



