24-6 



Mr. P. Nicholson on derivative Analijsis. 



Put B = <7, C = fl% D = fl' &c. and the operation will be 

 1+B.r+ C^''+ D^J +&C. 



1 +Bj- + Gr^+ Rf'+^cT 

 + b\t^ +&C. 



+&C. 



1+B,x+C.a,'^+ D.jtH&c. 

 Where B, = B + 6, 



C. = C+^.B + 6= = C + ^.(B + &) = C+iB, 



D, = D+^.C + ^.^B+^/^ = D + 6(C+^B+<^^-) = I>+^'C, 



&-C. _ &c. 



In the same manner by taking the series 14-B,.r*+C,A'* + 

 D,.r' + &C. as a multiplicand and the series 1 + cx-\-c'^x^ +c^x^ + 

 &c, as a multiplier ; then if the entire product be l+B^xH- 

 C .r* + D.„r' + &c. we shall have by the same law 



"B,= B,+C 



C,= C,+cB,+c' = C.+cB3 



D,=D.+cC,+c^B. + c' = r),+c(C.+cB. + cO = D.+cC, 



<S:c. &c. 



and so on for the product of any number of series ; therefore, ar- 

 ranging these values according to the number of products, there 



will arise Bj = B + i 

 B=B,+c 

 B, = B,+rf 



&c. 



C-C, + di, D„=D, + cC, 

 C,= C, + ^B, D3=D,+^C, &c. 

 &c. &c. 



Let it be required to find all the combinations of the letters 

 o, b, c, equally with one another to the third order. 

 Now observing that B=fl, C = a% D = a', then will 

 B,=«-f6 

 1st order. B, = 1- r 



2d order. C,=- 



C, = a^ + a6 + ^^ 



•+ac + hc + c'^ 



'Di = a^ + a'-b + ab'- + b^ 

 3d order. D, = \-a'^c+abc-\-b''c + ac' + bc'-+c^ 



"^ &3: 



Where the long line stands for all the combinations of the next 

 line above it. 



Again, let it be required to find all the orders of the com- 

 binations of the letters aaa, bb, c, or a', b^, c, or let all the di- 

 visors of 360 be requii-ed; now 360 = 2K3^.5=a^b^c. 



Here B, = a-|-6 

 1 st order. B . = h c 



2d order. C, = - 



C, = ff^ + «^>+6* 



• + «r + /;c for r'' is not wanted 



