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XVIII. — On the Approximation to the Roots of Cubic Equations by help of 

 Recurring Chain- Fractions. By Edward Sang. 



(Read 7th January 1884. ) 



In the twenty-ninth volume of the Society's Transactions, at page 59, Lord 

 Brouncker's process for finding the ratio of two quantities (commonly known 

 as the method of continued fractions) is extended to the comparison of three or 

 more magnitudes. It is there shown that recurrence, which was believed to 

 belong exclusively to equations of the second degree, extends to those of higher 

 orders, and examples of this extension are given in determining the proportions 

 of the heptagon and enneagon. 



In the present paper it is proposed to show the application of this extended 

 method to equations of the third degree. 



If there be a progression of numbers A,B,C,D,E,.... formed by 

 means of the multipliers p , q , r , according to the scheme : — 



rB+qC+pT)=E 

 rC+qD+pE=F, 



and if the number p be greater than either q or r, the terms will approach to 

 be in continued proportion, and their ultimate ratio will be the positive root of 

 the equation 



x z — px 2 — qx — r = , ..... (1) 



independently of the values assumed for the initial A , B , C . The actual pro- 

 gression may be regarded as the sum of three series having the initials A, 0, 0; 

 , B , ; and , , C respectively. On developing the term, we find that the 

 coefficient of A in the n\h term is r times that of C in the preceding or n—lst 

 term ; while the coefficient of B is compounded of q times the n — 1st, and r 

 times the n — 2d coefficients of C. Hence we need only to compute the series 

 beginning with , , 1 , in order to have the means of compounding any term 

 of a progression formed with the same multipliers. 



VOL. XXXII. PART II. 3 F 



