354 Mr. P. Nicholson on derivative Analysis. 



Subtract as many of the first cohimn as are opposite to 

 each of the other coknnns, from each of these other columns 

 will give each respective column of differences. 



Here A=l, B = 0, C = &c. 







_ 6x+4=-24|-24x— 2=48i48=E, 

 ]0x-2= + 20l- 4=D4 

 16=C. 



lX+2= + 2| + 2x-3 = -6 

 lX-3=-3 -lx+4 = -4 

 lX+4 = + 4 +3x-2=-6 

 lX-2=-2H-l=B' 



Whence (x+2)(x-3)(j:+4)(x— 2)=j:^4-J^^-16jr^-4cr+48 

 Transform 3(cr + 6)(;r+l)(^+4)(jr— 3) into the series 



+1 : +4 

 +4 : —3 

 -3 : 



















+ 18x+4=+ 721+ 72x-3=-216|-216=E 



+72x -4= -288l-216=D4 



-48=C4 



3x+6=:4-18|18x+l = -|- 18 

 3x+0=+ 0|l8x+3=+ 54 

 3x+2=i- 6|24x— 5=-120 

 3x-6=-18| 6=3^ 



Whence 

 S{x + 6){x-\-\){x + i){x — 3) is transformed to the series 

 Sx{x-\-\){x + 2){x+3)-^6x{x+\){x+2)-^8x[x+\)-'2\6x-^\6 

 Transform the series 5x* — 2x^-\-5x'^ — ^x-\-6 into the series 



5(x+l )(x+2)(:r + 3)(^+4)+B,(x+l )(^ + 2)(.r+3)+C,(a'+ 1 ) 

 {^ + 2)+D,(x4l)+E, 



: i 

 : : 

 : 



-2 



+ 5 



—4 



12x — l = -12l- 16x-l = 16|22=E4 

 46x -2=-921-108=D' 

 142=C4 



5x-l = - 51- 7x-l= 7 

 5x-2==-10 -17x-2=34 

 6x-3=-15 -32x-3=96 

 5x-4 = -20!-52 =B 



Whence 



5x^ — 2x^-\-5x'^ — 4j: + 6 is transformed to the series 



5(x+l)(^+2)(a:+3)(^+4)-52(^+l)(:r + 2)(^ + 3) + 142(^+l) 



(x+2)-. 108(^+1)4-22. 



Let it be required to transform the biquadratic function 

 x*-\-Sx^+ 2x'' ■\-^x + 5 to the form x{x+ \){x-^'2,){x+'d) + 

 B,x(x+1 )(^+2) + C,atr + 1 ) + D,a: + E,. 



Here A = (), B = 3, C = 2, D = 4, and E = 5. 



