Mr Grace, On the Zeros of a Polynomial. 355 



Consequently by the result just proved f'(z) is apolar to the 

 form 



+ J . »-i)C.-^»-») ^l- ( ^-^....o, 



or to 



(a n - 8 n ) - nz (a*- 1 - 8 n ~') + ?l ^ -1 ) ^ ( a n-« _ £«- 2 ) . . . = 0, 



that is to the form of degree w — 1, 



(a - *)« - (/3 - z ) n . 



Thus if/(V) = has two given roots a and /3, then/'(V) is 

 apolar to the form (a — z) n — (8 — z) n . 



(7) Now the roots of the equation 



(a - z) n -(8-z) n = 



are easily constructed, for they are given by 



(a-z) = <o(8- z), 



where <w is any ?ith root of unity which is not unity itself. 



To construct the points representing these roots take A, B for 

 the points a, 8 and let be the middle point of A, B, then 

 we have 



2j"7T 



Z — CL —t , „ - 



— - — =e n , r=l, % ...Ti-1; 



a + /3 



= iCOt 



' ' a-8 w ' 



2 



a+ 8 , rir a — 8 



■■■*--s- + ,00 *T-2-' 



which shews that to arrive at we have to travel a distance 



OA cot — 

 w 



from along a line at right angles to A B. 



The greatest distance of one of these (n — 1) points from is 



OA cot - , 

 n 



VOL. XI. pt. v. 26 



