350 MR. WARREN ON GEOMETRICAL REPRESENTATION 



c 

 34. Let a be a quantity inclined to unity at an angle less than -^, where c 



is the circumference of the circle ; 



then »■ = 2 {1^ + ^^y + i C-^y + &c.} . 



For let a = h /1\7, where i is a positive quantity, and — positive and less 



than :|, 



then d = V + a J — 1 ; 







a, 



Now, since a^=h /1\t, we have (by Treatise, Art. 135.) 



(!) 



(B + « v' - 1)'^^ 



(ay = 1 + (B + «y-^) X + ^"^%"--^- + &c. 

 where x is any possible quantity, 



,„aB=2{|^ + i(l^y + i(^y+&c.j 



= possible hyperbolic logarithm of h 

 = h'; 



o 



Also (by Treatise, Art. 132.) 

 /a\ = 1 + A aH- T^ + &c., 



where A = 2 {|^ + ^ (^)^ + i C-^)^ + &c.} , 



.•. equating the coefficients, 

 A = B + a J^^ 

 = ft' + « J^^ 



a'. 



35. Let a be a quantity inclined to unity at an angle less than -r, and let 

 a — 1 be in length less than unity ; 

 Then a' = a - 1 - J (a - 1)^ + i (a - 1)' - &c. 



