156 



R. FU.JTSAWA. 



(6) [P„+ a: + . . . +p,„+ir"'^^] + [a+ Qi:+ • ■ +9.-ir^M''^=o. 



Theii Ç2ui-{-'2) roots of tlie erjuntion 



are C, and r repeated (2m + 1) time.s. Thus we ohtain the followirig 

 identity : 



(7) {P,+P,Z + ... + P„,,,Z''^-^'}'^{Q, + Q,Z + ... + Q,,_,Z^>'''}\^^^ 



^PUAZ-:){Z-zr^\ 



Herein putting Z = 0, Z — \^ Z = ^7j- successively and reducing, we 

 obtain 



(8) 



V: 



Vi-r {i-z) ' 



Pn 



Vl-A-C {l-'k-z) 2 



p ' 



Po + Pi+...+P^^l ^ 



Pmfl 



P, 



jre+i 



In extracting squnre root, strictly speaking, we have to ])refix the 

 double sign ± ; but by taking some particular value of w or by putting 

 /,•=() in the final results to be hereafter obtained, it comes out that we 



have to take the + sign. 



§ - 



Let us now investigate the êX])ressions which occur on the right- 

 hand side of equation (<S). For this purpose, put 



