On extracting the Roots of Equations. 175 



Place the coefficients and absolute numbers A, B, C, &c. of 

 the given lunction, with their proper signs, in a line, and in 

 the same order as they stand in that given function. 



Add the product Ae of the first coefficient A, and the 

 quantity e to the second coefficient B, and write the sum a 

 under the second coefficient. 



Multiply this sum «' by the quantity e, and add the product 

 to the third coefficient C, and write the sum a" under C, the 

 third coefficient in a second line. 



Pi-oceed in this manner until the remaining coefficients and 

 the absolute number have been used, so that each respective 

 sum may fall in a line below that last found under the pre- 

 ceding coefficient. 



To complete the columns under the coefficient of the second 

 term, add the product of the first coefficient A and the quan- 

 tity e to the number a' found under the second coefficient, and 

 write the sum b' under this number in a line with the num- 

 ber a" under the third coefficient. 



Proceed with every succeeding quantity in the second column 

 in the same manner, until as many quantities or numbers have 

 been found as the exponent of the highest power contains units. 



In general, when any two adjacent numbers in the same 

 line or horizontal row are found, by adding the product of the 

 first number and the quantity ^to the second number, the sum is 

 the next number under the second of these two given numbers. 



Then, when all the columns are completed, write the coeffi- 

 cient of the first term oi' the given function before the number 

 at the bottom of the column under the coefficient of second term 

 of the proposed equation : then the lowermost horizontal row 

 will contain the coefficients of the transformed function. 



The form of oj)eration as now directed for the quadratic 

 function Ax' + Bx + C is as follows: 



X, A + B+ C {c 



X— r, A 4- // 4" «" 



So Uiat the function Ajr^ + Bo: + C is transformed to A(.r— eY 

 + b\x-c)ta". 



'J'he form of operation for the cubic function A.r*+B.r' + 

 C jr + D is as follows : 



A + B + C -f- D (<-- 

 + «' 



+ // + «" 

 h+ c' + h"+a "'. 

 And thus the cubic function A.r'+ B.r' + C.r + D is tnnis- 

 fonncd lo \{x-i-y-{-i',{x-cy-\-l,"{x—r)-\-a". 

 And so on lor tlic lii^^'licr powers. 



Ex- 



