m 



Ax 



On exiraciing the Roots of Equations. 



+ Bx""' + Cx""%&c.= 



|Pj;"+Q/"' +R/'^ +&C. 



Now this is the second equation of condition ; therefore 

 P'= P, Q'= Ve + Q, R'= Qf + R, S'= R^ + S, and so on ; 

 whence the proposition is manifest. 



Corol. 1. — Hence the whole function P'u" + Q'u ~ +RV" 

 -i-&c. may easily be derived from the given function 



Corol. 2. — Hence because P' = P, P in the firs, derived func- 

 tion must be equal to A, as will appear by the followhig table: 

 A = A 

 A.r + B = Au H-jB 

 Aj;"+Bx + C = Au^ + oBt' +2C 

 A.r^ + B.t'^ + C.r + D = Ai^ + sBt;^+ aCr +3D 

 &c. &c. 



Whence by the property shown in the proposition demon- 

 strated, the values of the coefiicients jB, o)i, 3B, &c. of the se- 

 cond term respectively of the first, second, third, &c. derived 

 function, the values „C, gC, &c. of the third term respectively 

 of the second, third, &c. derived function, and so on, may 

 easily be obtained from one another, as will appear by the fol- 

 lowing table : 

 jB = A^+ B 

 2B = Ac+iB ,C = ,B^+ C 



^ 3B = A(^+,B ,0 = 2^^ + 2^ 3D = X^+D 



&c. &c. &c. &c. 



And generally 

 „B=Ac+ „-iB,„C=„_,B^+ „_iC, „D = „_,Cf + „_iD,&c. 



Problem. — To transform any rational function of an im- 

 known quantity into another in which the unknown quantity 

 shall be greater or less than that in the proj)o.sed function by 

 a given difference. 



To perform this problem, it would only be necessary to at- 

 tend to the table at the end of the demonstration ; but to give 

 greater facility in the execution I shall endeavour to express 

 the table in words. 



Let (' rojircsent the c]uantity to be added or taken away 

 from the unknown (juantity in the pro2)oscd function, as in the 

 proposition. 



Place 



