322 EDWARD SANG ON THE APPROXIMATION TO THE ROOTS OF 



S3 1 y2 a 8 + 2qra 2 /3 +pr a 2 y + ( pr + cf) a ft 2 + (pq- 3r)a/3y - qay 2 + (pq + ^W 

 + (p 2 -qWy-2pPy* + y*} 

 -S 2 {A[3r 2 a 2 + 4qra/3 + 2pray + (pr+q 2 )p 2 + (pq-or)/By-qy*] 



+ B[2qr a 2 + (2pr + 2 2 2 )a/3 + (pq - 3r)ay + (3pq + 3r)/3 2 + (2f - 2q)fiy - 2py*\ 

 + C[pra 2 + (pq-3r)a/3-2qay + (p 2 -q)/3 2 -4,p/3y + 3y 2 ]} 



+ S{a[3r 2 A 2 + 4:qrAB + 22}rAC + (q 2 +pr)'B 2 + (pq-3r)BC-qC 2 ] 



+ 8[2qrA 2 +(2pr+2q 2 )AB + (pq-3r)AC+(3pq + 3r)B 2 + (2p' i -2q)BC-2pC 2 ] 

 + y[prA 2 + (pq - 3r)AB -2qAC + (p 2 - q)B 2 - 4pBC + 3C 2 ] } 



- S°{?- 2 A 3 + 2qrA 2 B +pr A 2 C + (pr + q 2 )AB 2 + (pq - 3r) ABC - 9 AC 2 + (pq + r)B* + 



(p 2 -q)B*C-2pBC 2 + C*} = (3) 



Thus it appears that, while the root of every cubic equation may be reached 

 by help of a recurring chain-fraction of the third order, every such fraction has 

 for. its asymptote the root of a cubic. The above equation (3) gives us directly 

 the form of the cubic when the initials and the scheme of progression are 

 known ; and, inversely, it contains the means for discovering the progression 

 suiting a proposed cubic. Thus for such an equation as 



GS 3 -HS 2 + KS-L=0, 



we must equate the above coefficients to G , H , K , L respectively. Here, 

 among the nine unknowns, p , q , r ; A , B , C ; a , fi, y , we have only four 

 conditions, so that we are at liberty to make five arbitrary assumptions. Now 

 of the six, A, B, C; a, /3, y , the third power of each occurs ; hence the 

 ultimate equation must contain at least one cube. Thus we are again thrown 

 back on the solution of a cubic ; but in this case we know that it is always 

 possible so to make the assumptions as that the root may be integer, provided 

 the coefficients of the given equation be so. 



The preceding very involved expressions may be replaced by others con- 

 siderably simpler. The nth. term of the progression may be written 



D[w] + E[n-l]+F[w-2] 

 d[n] + e[n-l] + F[n-2] ' 



where D takes the place of rB , E that of (rA + qB) , and F that of C ; and 

 similarly for the denominator. The asymptote then is 



whence 



(D~dS)a; 2 + (E cS)x-(Y-f)x=0 . 



