Application of the Fluxional Ratio, fyc. 10 J 



In the equation (3) p is considered as a function of x. Letp be 

 represented by /'a:, and p' by f'x, &;c. Inasmuch as the develop- 

 ments (2) (3) are identical, the terms which contain the same pow- 

 ers of h are necessarily equal ; consequently if we compare the 

 termsof these two series affected with ih, ih^,ih^, he. we shall find that 

 2g'=^'=/'^a;, hence q=\p'^=-2f"x. Again, 37'~q' = ^f"'r, hence 

 r = iq'=i.lf'''x. Again, 45=7^= 1.1/'% hence s = i?-^=i. J. i/^ar. 

 Again, 5i=s'=i.i.i/^a', hence i = is' =1.1.1.}/^ a,-. By thus 

 proceeding we shall find successively all the other coefficients of the 

 equation (1). Substituting in this equation the valuesof /?, 5-, r,s,hc. 



we shall h2.vef{x+h)=fx+fxh-{-f'x-^+f'"x^^^f-'xj^^-^ 



h' 

 f^'X cy cy 4 r -i-hc. (4.) This series is of a very general nature, com- 

 prehending that of a power of a common binomial quantity indicated- 

 by {x-\-hY y that of a power of a logarithmic binomial quantity indi- 

 cated by log. {x-{-hy, and that of a power of a binomial sine or co- 

 sine indicated by sin.(a;-f ^j", or cos.(^4-/0"- I" ^^^ equation (4.) 

 f'x, being the coefficient of A in the development of the common bi- 

 nomial quantity, is indicated by a^,-""' ; f'x, being the coefficient of 



'h^ . . . h^ 



-^-j is indicated by /3x'" ^ ; f"^^, being the coefficient of— -^-^ '1% in- 

 dicated by yx"'^, and so on : hence putting these values oi f'x, f'x,. 

 f"'x, fee. in the equation (4), we have /(.r+A)=a;"-f k.t"~'^-[- 



h^ h^ h' 



/3a;" 2+7^" ^"S~o"+<^^"~''o"Tr4 + &'C. (5.) which is the common 



binomial series. 



The several orders of fluxions are produced by multiplying their 



fluxional ratios, each, into the preceding order, that is to say, 



nx- 

 a^^X— 7- =n:r"~^r, the first fluxion. (6.) 



[n — l)x' 

 nx^-'x'X =n{n — l)x'"~^x'^, the second fluxion. (7.) 



(n-2)x- 

 n{n—l)x''-^x-^-X- :: — n{n-l)l{n~2)x"-^x'^, the third 



fluxion. (8.) 



Now if we consider equation (5.) we shall perceive that the coef- 

 ficient ot h, in the second term in the development of the binomial 

 (ar+^O^jistt-'c""' , and. is the same with the fluxional coefficient n^r"-> 



