OPERATIVE DIVISION 227 



Now write the equation in the form 



x — cx n = 1 



and find the form of by operative division. Suppose first 



— co n 



that n = 2. Then 



— co 2 ] [0 + co* + 2c*o s + 5c J o 4 + etc. 



O — CO 2 



5c s o 4 — etc. 



Applying this quotient to the absolute term, unity, we have 

 for the lesser positive root. 



x= 1 + c+ 2c*+ 5c 3 + 14c 4 + 42c* + i32c 8 + . . .; 

 = — ( 1 — \/i —4c) ; 



2C X ^ ' 



as shown by expanding the radicle by the binomial theorem. 

 The series is divergent if c> -, and the root is then unreal. 



If c = -, the roots are equal, but the convergence is slow. 

 4 



For the cubic x— cx 3 = 1 we have by the same method, 



x= [0 + co'-j- 3c V+ . . .]i 

 = 1 + c+ 3c 2 + i2c'+ 55c 4 + 273c 6 + i428c"+ • ■ • 



The series is divergent and the positive roots are unreal 

 if — ; and these roots are equal, but the convergence is 



slow, if c = — . 

 27 



For the general case x — cx n = 1 , on carrying out the opera- 

 tive division, we shall easily see that the successive coefficients 

 in the quotient are reducible to a simple binomial form ; and 

 we have 



T t m c* 



[o-coT 1 = o--(-«W , + — — (i-2w) V 2 "" 1 - 



L J W V ; 2W -I V ' 2! 



I , x (3) c J 



3» 



— (2-3») ^o 3 ' l - 2 +etc, 



~2 K J ' 3l 



