348 MR. WARREN ON GEOMETRICAL REPRESENTATION 



but ^ = possible hyperbolic logarithm of g, 







n = h, 



o 



... ^ + A' = . gh 







.-. d+h'^gN + a + ^+p + q.C.J-l; 

 P g 



but, since g h . (\\ c =:f, 



(g h\' + a"~+~^ J^^=if, if « + ^ be less than c, 



=/', not less - - , 



•'• ('^ ^Y + a + |3 + j? + g . c . ^ — 1 = /' , if a -f |3 be less than c, 



^ f , not less - - ; 



p + i + i 



.-. d ■\- V z= f J if a + |3 be less than c, 



P <l P + 9 



= /' , not less - - . 



P + 9 + l 



30. Cor.) Hence if a and b be any quantities whatever, and a be inclined 

 to unity at an angle = a, and b at an angle = (3, a and /3 being each positive 

 and less than c the circumference of the circle, and -j =f; 



then a' — b' = f , if a be not less than j3, 



P 9 P-1 



= f , if a be less than |3. 



31. Let a and b be any quantities whatever, and let a be inclined to unity 

 at an angle = a and b, at an angle = (i, a and |3 being each positive and less 

 than c the circumference of the circle, and let a b =f, and let ?n be any 

 quantity whatever ; 



then /aV" . (b^ = / / N"*, if a + |3 be less than c 

 = ( f \^> ^^^ l^ss - - • 



