I 



Mr. P. Nicholson 072 derivative Aiialj/sis. Si? 



D, = «^+a*6+«i^ for b^ is not wanted 

 3d order. D,= -[- a^c ■^- abc + b-c 



^^ — a^li-^a^b'- no higher than a' or i^ being wanted 

 ith order. Ea= ■ -\-a^bc+a--b'^c 



l\ = a^b'' 



5th order. F,= +n^bc-{-a-b'c 



G, = o 

 6th order, Gj = a ' i ^c 



Parf IT. — Division. 

 Division is performed in the usual way, viz. by arranging 

 the parts of the dividend in a line according to the natural or 

 inverse order of the powers of the variable, and the paits of 

 the divisor in the same order. 



Divide the first part of the dividend by the first part of the 

 divisor, and the result is the first part of die quotient. 



Multiply the first part of the quotient successively, by every 

 part of the divisor, and place the products so that the powers of 

 the variable may be under the same powers in the dividend. 



Draw a line underneath and write in a line below the line 

 thus drawn the same powers of die variable as those imme- 

 diately above, except in the first place, and prefix a new let- 

 ter to each power as a coefficient which will form the first re- 

 mainder. 



Annex the next part of the dividend from which no sub- 

 traction has been made to this remainder, and consider this 

 remainder so increased as a second dividend ; then proceed to 

 find the second or next part of the quotient, and the third or 

 next dividend as before; and so on, as far as may be neces- 

 sary. In any convenient place write the letters thus substi- 

 tuted, and their values, in the form of equations; that is, every 

 letter equal to the aggregate of the two coefficients of the coi- 

 responding power above, considering the sign of die lower of 

 these two changed by subtraction. 



Then the table thus formed will show the law of derivation 

 by which the real quotient may be obtained. 



Ex. 1 . Divide « + ^x+yx^ + S.r' + &c. by 1 - bx put A = « ; 

 then proceed with die operation, 



Dividnul. Divisor. 



A + ^x + yx'-tdx^+SiC. \l-bx 

 A — Abx quotient 



B.:+y.t- I A+B.r-LCr'-+&C. 



Bx—Bbx"^ 



Ct^Cbx^ 



D.r'i-&c. 



&c. By 



