REPORT ON CERTAIN BRANCHES OF ANALYSIS. 319 



algebraical resolution of an equation whose roots can be repre- 

 sented by 



X, 6 X, &^ X, . . . . 6^-' X, 



where &^ x = x, and where 6 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, 

 if 9 ^ and d^ x express any other two of the roots, we have like- 

 wise 



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. 



o -J. 



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

 /*■ 

 cos 2 a, cos 3 «, . . . cos /x a is 



^^_|.^^-2 + ^./i^ii:_l)^^-4. .. =0 (1.) 



which may be easily shown to possess the required form and 

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

 is, as is well known, a rational function of cos « or a:* ; and, 

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

 likewise 9 9i a: = cos mm-^a =■ cos ^Wj m a = 9i 9 x, which is the 

 second condition which was required to be fulfilled. 



Let us suppose /«, = 2 « + 1, when the roots of the equation 

 (1.) will be 



Stt 47r 4<nit ^ 



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

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

 one o{ n dimensions, which is 



x" + ^ x"-' - -T- (« — 1) x"-2 — — (n — 2) x"-3 



1 (n-2){n-3) l_ (»-3)(/»-4) _ 



+ 16' 1.2 "" ^ 32' 1.2 "" i^c.-U{4.) 



whose roots are 



2 TT 4 TT 2 M 



cos ;z r, cos p: — ^, .... COS 



TT 



2n + I 2/2 + 1 2« + 1 



