REPORT ON CERTAm BRANCHES OF ANALYSIS. 319 



algebraical resolution of an equation whose roots can be repre- 

 sented by 



x,Qx,&'^x, . . . . r-'a;, 



where S'* x = x, and where 9 is a rational function of x and of 

 known quantities ; and also of an equation where all the roots 

 can be expressed rationally in terms of one of them, and where, 

 ifdx and flj x express any other two of the roots, we have like- 

 wise 



6 Q^x = QyS X. 



It is impossible, however, within a space much less than that 

 of the memoir itself, to give any intelligible account of the pro- 

 cess followed in the demonstration of these propositions, and 

 of many others which are connected with them. We shall con- 

 tent ourselves, therefore, with a slight notice of their applica- 

 tion to circular functions. 



If we suppose a = — , the equation whose roots are cos a, 



cos 2 a, cos S a, . . . cos j* a is 



;,^_|.^^-2 + ^./fL^i_:^)^-4. .. =0 (1.) 



which may be easily shown to possess the required form and 

 properties ; — for, in the first place, cos m a =■ ^ (cos a), where 9 

 is, as is well known, a rational function of cos a or ^ ; and, 

 in the second place, if 9 a; = cos m a and 9, a; = cos m^ a, then 

 likewise 9 9^ a;* = cos mm^^a ■=■ cos m^ ma = d^d x, which is the 

 second condition which was required to be fulfilled. 



Let us suppose [^ =■ 2n + 1, when the roots of the equation 

 (1.) will be 



27r 47r 4w7r _ 



*=°^ 2irrr ^°' 2¥TT • • • "''' 2;rrv "^' ^ "' 



of which the last is 1, and the n first of the remainder equal to 

 the n last. The equation (1.) may be depressed, therefore, to 

 one of n dimensions, which is 



x" + ^ x"-'' - ^ (w - 1) x"-"" --L{n — 2) x"-" 



1 (n-2){n-3) 1_ {n-3){n-^) 



+ I6' rT2 "" +32* 1.2 '^. i^c.-U{^.) 



whose roots are 



Stt 47r 2 WTT 



