Mr. P. Nicholson on derivative Analysis. 245 



has coefficients wiU be formed, which will show the relation 

 between the succeeding and the preceding coefficients of the 

 entire product, or between the coefficients of the entire pro- 

 duct and those of the multiplicand and multiplier. 



Ex. 1. Multiply A+B^' + C^^+D^^+E^^+&c. by the 



binomial a-\-x. 



Operatiott. 



A+ B^+ Cx^+ D^'+ Ejr*+&c. 



a-\- X 



a A +aBx -\-aCx'-{-aDx^+aY.x''-\-hc. 

 A.r+ Bx^-\- C^'+ D^r-t-f-Scc. 

 A,4- B,x+C.x^ +D,x'+ E,/c++&c. 

 From which we have the following derivative equations, viz. 



A, = aA 



B, = A+aB 



C, = B +flC 



Hence it appears that the entire product may be derived 

 from the multiplicand, and the constant part of the multiplier. 

 Since any coefficient of the entire product is equal to the par- 

 tial product of the corresponding coefficient of the multipli- 

 cand, and the constant part of the multiplier plus the pre- 

 ceding coefficient of the multiplicand. 



Ex.2. Multiply x"-i+B^'-2+Cx"-34-Dx-*-hEa7"^+&c. by 

 the binomial x-ra. 



x»-i+ Bx"-2^ Cx'^Jf D,r"-*+ E.r"-5+&c. 



■r + a 



x" + Bx"-^+ Ca;"-24- T>x^-^ Ex"-*+&c. 

 _|_ ax^'^ +«B^"'+aCx"-^+flD^"- *+&c. 



Whence we have the following derivative equations, viz. 

 B.= B + a 

 C,= C+cB 

 D.= D+gC 

 E. = E + aD 

 &c. 

 From which it appears that the entire product may be de- 

 rived from the multiplicand; for the coefficient of any term of 

 the entire product is equal to the coefficient of the correspond- 

 ing term of the multiplicand plus the partial product of the 

 preceding term of the multiplicand, and the second part of 

 the multiplier. 



Ex. 3. Multiply the series I4-ax+a\r"-f-«'x'-|-&c. by the 

 series \-\-bx-\-b'x^-\-b^x^-\-^Q. 



Put 



