Mr. P. Nicholson o« derivative Atialysis. 249 



But if the first part a of the divisor were unity, the deriva- 

 tive table would be simply 



B =cib-\-^ 



B.=«c + 7 C=BZ) + B, 



C. = Bc+ 8 D =C6+C, 



&c. I&c. 



And the quotient derived from this table would be simply 

 «+B.r+ar^ + &c.= 



a + («i + /3> + (a6^ + /3& + «C + y)a:- + &C. 



Ex. 3. Divide the series a + /3.r + y.r--f 8x^ + ea' + &:c. by the 

 series l—hx— cx'^ — dx^ — ex-* — &.c. 



Operatio7i. 

 Dividend. Divisor. 



\\—bx— cx"^ — dx^— ex^ — &c. 

 « + /3t + yx"" + Ij^-{- sx'' +&C. 1 Quotient 

 a.—a.hx —a.cx"- — a.dx^ — ctex* -&c. |«-|-Bcr+C^^+Dj:'+&c. 

 Bx+ B^^+ B;r^+B,a:* + &c. 

 Bx-b^x^ -cB> -f/BT*-&c. 



Cj,'^ + C.r*+ Cx* + &c. 

 Cx'- -bCx^- cCx^ - &c. 



T>x^ + bVx^-S^c. 



Ex*-\-Sic. 

 Ex*-Sic. 



By this operation we have the following derivative equa- 

 tions, viz. 

 B =bu-j-i3 

 B, = r«+y 

 B. = dci+'6 

 B^=ea+£ 

 &c. 



C =&B+B, 



C, = cB-fB, 



C, = r/B+B, 



&c. 



E = iD+D. I 

 &c. &c. 



D. = cC+C, 



&c. 



By multiplying and adding as this table directs, we shall 

 have the real coefficients of the powers of r in the quotient, viz. 

 A=« 

 B=«6 4-/3 

 C=uh^+^b 4-«c-|-y 

 'D = ab^+^b^+acb+yb-\-ccbc-\-^c + cid+ 5 

 Ike. 

 Or, if the divisor had been a — bx—cx'^ — dx^ — Si.c, instead 

 of 1 — /j.r— r.r- — f/.r' — &c. and if A had been the first term of 

 Vol.()2. No. 306. Off. 182.S. I i the 



