CUBIC EQUATIONS BY HELP OF RECURRING CHAIN-FRACTIONS. 315 



of which the terms (3), (6), (9), (12) give, on being simplified, the pro- 

 gression 



(0) (1) (2) (3) (4) 



-2 44 66 658 100 986 738 152 994 708 140 

 1 ' 0' 23' 34 846' 52 791621' 79 979 201300' 



for which the multipliers are r=l , q— - 3 , and jp — 1515 . 



In order to get a clear view of the general principles here involved, we 

 shall propose to extract the cube root of n 3 + 1 . 



The equation a 3 — (n s + l)b 3 = Q , gives, for the first approximation, a = nb + c, 

 whence 



b 3 -3n 2 Wc-3nbc 2 -c 3 = 0, 



so that the multipliers are r = l , q = Sn, p = 3n 2 , which give, converging to the 

 ratio of b : c, the progression 



0, 0, 1, Ziv 1 , 9?i 4 +3tt, 27» 6 + 18w 3 + l, &c, 



and consequently, converging to y/(n s + l), the series of fractions 



(0) (1) (2) '(3) (4) 



_1_ 0_ n_ Sn 3 + 1 9n 5 + 6n 2 27w 7 + 27^* + 4n, 



' ' 1 ' 'in 2 ' 9n* + 3n ' 27» 6 + 18ti 3 + 1 ' 



(5) (6) 



81w 9 + 108ffl 6 + 33ffl 2 + l 2437^ 1 +40ow 8 +189to 5 +21w 2 

 81n s + 81n 5 + 15n 2 ' 243m 10 + 324% 7 -j-108^ + 6» ' 



(V) 

 729^+1 458rc 10 + 918^ 7 +189« 4 +7rc 

 729% 12 +1215?i 9 -|-594?i 6 +81#+1 ' 



(8) 

 2 187?i 15 +5 103ft l2 +4 050^+1 242^ g +H7ft 3 +l 

 2 187» M +4 374w n + 2 835% 8 +648» 5 + 36% 2 ' 



(9) 

 6 561-» 17 +17 496rc 14 +16 767ro"+6 885?t 8 +l 107rc 5 +45w 2 

 6 561n 6 + 15 309«, 13 +12 393w 10 +4 050» 7 +459w 4 +9?i ' 



Here we observe that the numerators of the terms (3), (6), (9) are divisible 



