358 MR. WARREN ON GEOMETRICAL REPRESENTATION 



58. Ex.) Let the series be I, 2, 4, &c., 



then r = 2, 



r> 2' 



and 1 = 1 ^ , 



.-. the values of l"" "^^ ' are 1, 2, 4, &c., also J, | &c. 

 59. Let it be required to find the values of ^ — 1 • 



1' = c y^^, 



■•■(0)0' =7^^' 



but V - 1 = (1)^ 

 .-. since -j is less than c, (^ — \\ = -^ >/ — 1, 



.-. ({'^Y^Y^^y =ii + p)c y^^^T. y-:ri = _ (i + ^) c, 



■•■ (^7 ') = (?) = (E)«+"" 



let 0, 1, 2, &c., also — 1, — 2, &c., be successively substituted for j9, then we 

 have the values of*/ — l"*^" ^ as follows, viz. 



1 1 1 



5 c 9 c 



(?)^'(!)^ '(?)*'«'-' 



3c ?£ 



4 



(?)H?) 



&C., 



all in geometric progression. 



60. From what has been demonstrated it will be manifest that all algebraic 

 quantities may be geometrically represented, both in length and direction, by 

 lines drawn in a given plane from a given point. 



