CUBIC EQUATIONS BY HELP OE RECURRING CHAIN-FRACTIONS. 325 

 The multipliers r=l , q= — 6 , p — 9 thus found give the progression 

 -1 3 26 216 1791 14 849 



' ' 1 ' 9 ' 75 ' 622 ' 5 157 

 which contains each alternate term of the preceding. 



, &c, 



The construction of a regular polygon of eleven sides involves an equation 

 of the fifth order, and would introduce chain-fractions also of that order. The 

 extension of the present method to that case offers no difficulty, but would 

 pass beyond the scope of this paper. 



In the preceding examples we have several times examined the progression 

 formed by each third term of the series ; and in the last example we have 

 noticed the progression of the alternate terms. This brings us to the general 

 law, that the terms taken at equal intervals along a series of recurring chain- 

 fractions form a series of the same kind. Thus [0] , [2] , [4] , [6] , &c, are 

 connected by the law 



[n-4:]{r*} x[it-2]{2pr-q 2 } + [n]{pi + 2q} = [n-2] , 



the multipliers being 



R = r 2 , Q = 2pr-q*, F=p 2 + 2q. 



And, similarly each third term forms a progression according to the law 



[n-6]{rS} + [n-3]{q 3 -3pqr-3r 2 }-[n]{p^ + 3pq + 3r} = [n + 3], 



where 



E = r 3 ; Q = g2-3p2r-3r 2 ; F=p 5 + 3pq+3r . 



In the same way, for the terms four steps apart, we have 



R=-r 4 ; Q=-2p 2 r 2 + 42?2 2 r-2 4 + 42r 2 ; 

 P =p* + Ap\ + Apr + 2g 2 . 



This law of recurrence extends to chain-fractions of all orders, and even to 

 periodic continued fractions. Thus, in seeking the square root of 7 by the usual 

 process, we get the successive quotients 2; 1, 1, 1, 4 ; 1, 1, 1, 4; &c, 

 occurring in groups of four, and giving the converging fractions, 



2111441114 1 1 1 



1 2 3 5 8.3_7 45 82 127. 590 717 1307 2 024 

 0' 1' 1' 2' 3' 14' 17' 31' 48' 223' 271' 494' 765' 



