Mr. P. Nicholson on derivative Analj/nis. 353 



■ Write the successive sum of the coefficient ot" the second 

 term of the given function and each product in another co- 

 lumn on the right, and the last of these successive sums op- 

 posite the last product is the coefficient of the second term of 

 the transformed function. 



From the fii'st, second, &;c. factors of the second term of 

 the given function subtract the fii'st, second, &c. factors of 

 the second term of the function of which the coefficients are 

 required each from each. 



Multiply each of the successive sums by each of the dif- 

 ferences of the factors of the second terms, and place the pro- 

 ducts in a column under the coefficient of the third term of 

 the given function. 



Write the successive sums of the coefficient of the third 

 term and the products in a column on tlie right, and the last 

 of these successive svuns is the coefficient of the third term of 

 the transformed function. 



Proceed in the same manner to the last coefficient, which 

 being added to the product under it, the sum will be the abso- 

 lute number of the transformed function. 



Transform the cubic function of binomial factors 

 {x+5){.r-3){x+2) + 3{x-3){x-{-2)-5{x+2)-\-6 into 

 {x-h3){x-\-2){x-5)+B,{x+3){x+2)+Cix+3)+~D, 



3-5 6 



+ 3 

 + 2 

 -5 



+ 5x -6=-30|-3ox-l=35|41=D; 

 j-Ox±0=+ 0|-35=C, 1 



+ 7 = B3 



+ 5 : -3 1+2 +2 

 — 3 : +2 : —5 



+2 : +7 



Whence * 



(.r + 5)(^--3)(.r + 2) + 3(.r-3)(^4-2)- 5(x+2)+ 6 is trans- 

 formed to 

 {x+3){x-\-2){x-5) + 1{x+3){x-h2)-35{x-\-3)-\-4^1 

 Transform {x + 2){x-3){x+i){x-2) in terms of the powers 

 of X. That is, to transform 



{a:-\-2){x-3Xx-\-i'){x-2) + 0{x-^3){x+i){x-2) + 0{x+4^) 

 (x-2) + 0(a--2) + into 

 {x + 0){x + 0){x + 0{{x + 0) + B,{x + 0){x + 0){x+0)+C,{x^-0) 

 {x-\-0) + (D,{x+0)-\-E, 



- ■ —2 



♦ The differences above are found mentally, as 5 minus 3 is +2, and 

 —3 minus 2 is —5, and 2 minus —5 is +7; these arc the respective dif- 

 ferences of the factors in the first terms. Again, —3 minus +3 is —7, 

 and +2 minus -f 2 is ; these are the differences of the factors of the se- 

 cond terms. Again, +2 minus +'.i is —1, which is the dltfcrencc of the 

 factors in the third term. 



Vol. G2. No.307. iVm'. 1823. Vv -Sub- 



