Application of the Fluxional Ratio, &fc. 95 



n{n—\)[n—2)x''-H'''y}- ^=rj(n— l)(n— 2)(n— 3)a;"-''a;-*, 



the fourth fluxion. 



Let x' be the increment of a;, then if %-{•%' be raised successively 

 to the several powers, the increment of 



x^ will be 2xx'+x"' 



a;3 3j;23./_^3^^/2^2./3 



a;4 ^xH'-\-^x''x'^-\-Axx"'-\-z'^ 



In any given power, suppose the first fluxion divided by the inde- 

 terminate quantity A, the second fluxion divided by the indeterminate 

 quantity B, the third by C, &:c. to be equal, each, to the correspond- 

 ing term in the increment, x' being supposed equal to %' ; then all the 

 orders of fluxions, taken until the variable quantity becomes constant, 

 will be equal to the whole increment, because all the parts taken to- 

 gether are equal to the whole. The values of these unknown coefii- 

 cients are found in the following manner ; suppose x-\-x' is raised to 

 a given power, for instance the third, then the increment will be Sx-i' 

 -\-Sxx''-{-x'^ ; the first fluxion will be 3x^x' ; the second fluxion, 



3a; 2 a;- 6xx'^ 6x-^ 

 6xx'' f and the third fluxion, 6r ^ ; then— r — +^d~ +T^ =3x^2;/ 



2x^x' 6xx-2 



+ 5xx^'+x'^. By supposition —^=2x^x', and — o~ =3xx''', 



Cx-^* Sx^x' 6xx-2 



and Q-=a;'% hence A=l, B=2, C=6 5 therefore — z — + — x- 



6a;- 3 

 -^-g — ^^=3a;2x^+3xx'2+2;'^j the increment. When x+x' is raised 



to the fourth power, the increment will be 4x'x'-{-6x-x'^+4xx'3 + 



4x^x' 12a;2x-2 24a;x-3 24x'* 

 x'*. Proceeding as before ~t~ -\- — 5 — -f p + t^ = 



4x^x'-\-6x'^x'^-\-4xx'^-{-x'K By supposition —x~=4x^x', hence 



12a;2x'? 24a:a;*3 



A=l ; g — =6a;2x'2 ^ hence B=2; — ^ =4xx'% hence C= 



^ 24a;-* 4x3:c- 12:^2^-2 



6',~Y^—=x'\ hence D=24. Therefore 



24xx'^ 24a;"* 



— 6 — -h~n7~='ix^x'-\-6x^x'^-^4xx'^-\-x"^,l\ie'mcrement. To 



avoid the difiiculty of indicating by points a fluxion of a very high 



