CUBIC EQUATION'S BY HELP OF RECURRING CHAIN-FRACTIONS, 

 produce the progression 



319 



5 ' 12' 



54 

 34 ' 



94ft 



414ft 1 150^ &c 

 261ft ' 724f ' 



converging rapidly to the value of 1/4. 



For the cube root of 5 the equations are 



a 3 - 0a%- Oab 2 - 5& 3 = 

 _ 4&3 + 3 £2 C+ 3hc 2 + lc s = 



+ 3c 3 - 3c 2 d- 9cd 2 - 4^ 3 = 

 — 10^ 3 + 15dh+ 15fZc 2 + 3e 3 =0 

 + 13e 3 - 45c 2 /- 45c/ 2 -10/ 3 = 

 -78/ 3 + 219/V + lll// + 13 /? 3 = 

 &c. 



a = lb + c 

 b=lc +d 

 c ='M + e 

 d=2e+f 



f = 3g + h 

 &c. 



The equation in d and e gives the multipliers 



r = .S; £=£1.5; j»==1.5; 

 while « = 5c?+2<? ; & = 3^+le ; hence for 1/5, we have the progression 



5 9.5 21.75 48.375 108.0375 241.14375 538.284375 



3' 5.5' 12.75' 28.275' 



63.1875 



141.01875 



314.791875 



&c. 



Here the error is reduced about 5 times at each successive step. 

 The equation in e and f gives 





1 690 

 13 3 



91 = 



585 

 13 2 



45 



13 ' 



while a = 12* + 5/; 6 = 7^ + 3/. 



The progression thence resulting is 



12 



605 



34 245 1 915 230 107 241 125 



7 ' 354 ' 20 025 ' 1 120 045 ' 62 714 910 

 converging more rapidly than the former. 



, &c, 



From these instances it is clear that the cube root of any number, or the 

 root of any cubic equation with integer coefficients, may be represented by a 

 series of chain-fractions of the third order ; and not by one only, but by many 

 of such series. Since the successive steps of the Brounckerian process neces- 

 sarily depends on the peculiarities of the case, it would be difficult to make a 

 general analysis beyond the first step ; but a symbolical investigation that far 

 may lead to important results. 



VOL. XXXII. PART II. 3 G 



