248 



Mr. P. Nicholson on derivative Analysis. 



By this operation we have the following derivative table, 

 viz. 



A= « 

 B=A6-|-/3 

 C = B6-f-y 



&c. 

 From which it appears that the wth coefficient of the quo- 

 tient is equal to the product of the next preceding coefficient, 

 and the coefficient of the second term of the divisor plus the 

 n\h coefficient of the dividend. Whence by the table we de- 

 rive the coefficients of the quotient thus, 

 A= «=« 

 B=A^> + /3 = «6 +^ 

 C = B6+y=«i*+/35 4-y 



&c. &c. 



Whence 



A-|-B.r+C^»+&c. = «+(«5+^)x+(«5^-|-/3J-|-y)x*-f&c. 



Ex. 2. Divide the infinite series a + /3x + yx* + Ix^ + ix^ + &c. 

 by a — bx—cx'^. 



Operation. 

 Dividend. Di\nsor. 



a+j8jr+ yx^ +S^' + f:r'* + &c. a—bx—cx'- 



Bx- 



Quotient 



a B C , 



— I jrH x^ 



-f &c. 



£2. ,1 £l 3 



Cx^'+C.x^ + ex* 



C, iC 3 cC ,0 

 x^ X'' .r^&c. 



a a 



Dcr* + D,.r*+&c. 

 &c. &c. 



From which operation we have the following derivative table, via; 



B = 

 B.= 



ac+ya 



c = 



B6+B,a 

 a 



Bc+Sa 



D = 



C6+C.n 



a 

 &C. 



From this table we derive the real quotient 



a ' a* ' ^ nJ ~ ' 



&C. 



But 



