250 



Mr. P. Nicholson o« derivative Analysis. 



the quotient, B, C, D, &c. the coefficients of the following 

 terms, we should have had 



A= ^ 



B = 

 C = 



D = 



aft-f/Sa 



cd)^-\-liab-\-aac-\-ya'^ 



a* 

 &C. &C. 



That is, by making up the sum of the indices to the same 

 number as the highest in each part of each numerator, and 

 making the denominators respectively a, a\ a^, o*, &c., so that 

 if we have the coefficients in one way, we can easily find them 

 in the other. 



But if «, Q, y, &c., and a, b, c, &c. had been given in num- 

 bers, the values of A, B, C, the coefficients of the quotient, 

 would have been found much more easily by the rule directed 

 in the table, as we shall have occasion to show hereafter. 



Ex. 4. Divide the series ax'" + /3.r'«-^+y.r'"-2+8j'»'-=' + &c.by 

 the series x" —bx"-^ — cx"-'^—dx"-^—hc. 



Divisor. 



— hx"-^ — i 



ax'^-\- ^x'^^ -\- yx^'-^-ir g.r'«-^+&c. 



jtx"'— bux'"'-'^ — cou c''"-"— docx^-^ -&c. 



"~ B.r"'-i+ B,x"'-2+ B^"'-^+ &c. 



B^'"-'— ^Bx"^^— cB.r'"-^— & c. 



C.r"'-^-&Cx»'-3-&c. 

 Dx"'-3+&c. 

 T>x'^^~8ic. 



Quotient. 

 ax"'-"-\-Bx'""-'^-\-Cx"'-''-^ -f &c. 



&c. 

 By this operation we have the following derivative equations, 

 B =i«-t-/3 



C=6B+B, 



Tf=cC+C, 

 &c. 



B,= c« + y 



Bj=^«-|-8 



B3=m+g 



&c. 



C,= cB-}-B, 



C,=^B+B, 



&c. 



E = bT> + T>, 



&c. 



&c. 



which are the same as those in the table of the precedino' ex- 

 ample. ° 



Ex. 5. 



