352 



Mr. P. Nicliolson on derivative Analysis. 



And by comparing the coefficients of the corresponding 

 terms of the second sides of equations 6 and 7 will be foimd 



Let now V ±=u-\-a, w =u+b, x =u-\-c &c. 



Also let i/=7<+a, '•dy'=M+/3, x'=u-{-y &c. 



Then will ai=a—u, b.,=b—u, Cy=c—a. 

 a.^ — b — ^, b^—c—^ &c. 

 a,=c—y, &c. 

 &c. 

 "Whence 

 B, = B+(a-a)A a=C+(^.-«)B. D,=D+(f-«)C, 

 B„ = B. + (6-^)A C3=C,+(c-^)B, &c. 



B, = B,+ (c-y)A &c. 



&c. 

 A very convenient form for the arithmetical operation is as 

 follows : 



For a simple function, 



B 



(«-«) I B, 

 For a quadratic function, 

 B C 



(a— «) A 

 {b-^, A 



B,x(6-«) = Q, I C. 

 B, 



B 



For a cubic function, 

 C 



D 





(a-«)A B,x(6-«)=y, 

 (i-/3)A B,x(c-/3) = Q, 

 (c-y)A Bj &c. 



Explanation. 

 B +(a— «) gives B,, and B, x (i — «) gives Q,, C + Q, gives C.^ 



B.+(6-/3) B, ... B,x(c-/3) ... Q., C,+Q,, ... C, 



B,+(c-r) Bj &c. 



&c. 

 The rule thus exhibited may easily be expressed in words 

 at lengdi as follows : 



Place the coefficients of the given function in a horizontal 

 line. 



From the first, second, third, &c. factors of the first term 

 of the given function, subtract the first, second, third, &,c. 

 factors of the first term of the function of which the coeffi- 

 cients are required each from each. 



Multiply the coefficient of the first term of the given func- 

 tion by each of the differences, and write the products in a 

 column under the second coeflicient. 



Write 



