CUBIC EQUATIONS BY HELP OF RECURRING CHAIN-FRACTIONS. 321 



3/L 

 And again, if we write K + a = L or a = L— K, \J i? results, with the 



multipliers, 



(L-K)«; 3(L-K>; 3L 2 +21KL + 3K 2 ; 



from the initials 



+ (L-K- 3 . 0. K + 2L , 2K 3 + 30K 2 L+42KL 2 +7L 3 

 -(L-K- 3 ' 0' 2K + L'' 7K 3 + 42K 2 L+30KL 2 +2L 3 ' 



These inquiries have been confined to the components of two terms 

 only of the elementary progression, whereas in chain-fractions of the third 

 order three terms are admissible. For the purpose then of giving the utmost 

 generality to our research we shall suppose the three initial terms of a progres- 

 sion to be 



ABC 



' ID ' ' 



a p y 



the multipliers being, as before, r, q, p. Then, according to what has been 

 already shown, the /zth subsequent terms is 



[w-2]rB + [M-l]{rA + gB} + |>]C 

 [n + 2]r/3 + [n-l]{ra + qp} + [n]y ' 



If then x be the asymptote of the elementary progression, while S is that of 

 the series of fractions, we must have 



rB + (r A + qB)x + Gc 2 _ g , , 9 , 



and we wish now to express S directly in terms of the nine data, A, B, C ; a, 

 fi> y ', P, q> r- For this purpose we must eliminate x from the two equations 

 (1) and (2). 



Equation (2) may be written in the form 



(C-yS)x 2 +{r(A-aS) + q(B-/3S)}x+r(B-l38)=0 (2) 



from which and 



x 3 — px 2 — qx— r=0 (1) 



we have to eliminate x. The elimination gives 



