Mr. p. Nicholson on derivative Analysis. 351 



B„=B„.,+a„A, C„=C^i + 6„B„.„ D„=D„., + c„C„., &c. 



For let V — u'=a,, 



to— 'Du'=aj, w—v—b^y 

 x—x'=a^y x—'w' = b.jf x—v'=C3y 

 &c. 

 Then by transposition 

 V = v -{-a, 

 TO= to' -\-a.2=x/-\~b.;i 

 X = x' ■i-a.^z='w'-\-b,=v'-\-c, 

 &c. 



(1) ■ 



Now since A = A 



Multiply the first side of this equation by v, add B to the 

 product and multiply the second side by t/+a, and add B to 

 the product, and /g) 



(3) 

 Let Av+B = Ad'+B, 



Multiply the first side of this equation by xv, add C to tlie 

 product, and of the two terms on the second side multiply 

 the first by w'+a^, the second by v' + b, and add C to the 

 product, and /^\ 



A . n . r- (Ax/w'+ B,v-h C 

 At.c+Bt.+C=| +aAv'+b.,B, 



(5) 

 Let Avxo + Bw+C = Av'w + B.v' + C^ 



Multiply the first side of this equation by x, add D to the 

 product, and of the three terms on the second side multiply 

 the first by x' + a^, the second by 'w'-\-bj, the third by v'-i-c^, 

 and add D to the product, and 



A I -o . r> , T\ ( Av'w'x'+ B.yVw'-}- Cov'+ D 

 ATmr4-BTar+C.'+ D= | +aAv'w' + b^B]v'+c,C, 



(V) 

 Let Avwx+Bwx+Cx+'D=Av'w'x'+Bjv'w'-\~ Cy+ Dj 



&c. &c. 



Now, by comparing the coefficients of the corresponding 

 terms of the second sides of equations 2 and 3 will be found 



B, = B + a,A. 

 And by comjiariug the coefficients of tlic corresponding term 

 of the second sides of e(|uations 4- and 5 will be found 

 B,= B.+«,A, C,= C+A,B.. 



And 



