96 Application of the Fluxional Ratio, 8fc. 



order, when we wish to express it generally, x may be written [»J- j 



and X may be written [x^^ and so on. Suppose that E represents 

 any power cc" generally, the fluxions of the several orders are express- 

 ed in the following manner. In the preceding case n stands for the 

 exponent 4, and E stands for x^ ; [E]' for 4x^x-, the first fluxion ; 

 [E]2 for 12a;2a:-2 the second fluxion; [E]^ for 24a?a:;-Mhe third 

 fluxion; [E]* for 24a;'* the fourth fluxion; hence 4x^x'-\- 



-^-+-T-+-24-=tE]^+^-/ +4-+^=incre- 



ment. The larger the exponent n is taken, the greater will be the 

 number of terms, of which the series is composed. When n is in- 

 definitely large, the series becomes infinite, and in that case E stands 

 for a"; [E]> for nx^'-'x-; [E]^ for n{n — l)x"-''x'^ ; [E]=' for 

 n{n—l){n—2)x"-^x'^', [E]* for n{n—l){n—2){n—S)x"-*X''", 

 \Ey for n{n — l)(n — 2)(n — 3)(w — 4)x''-^x'^ ; Sic. The series 

 expressing the orders of fluxions becomes ( 1 .) [E] ' + [E] ^ + [E] ^ + 

 [E]* + [E]* + [E]8-l-&;c.... in inf. and the series expressing the 

 increment becomes 



(2.) (a;+:.0"-^"=[E]'+H^+E|^+El+_E^ 



in inf.=increment. 



The series of Maclaurin is 



FE]' rE]=* \Ey [EV 



2.3.4.$z-*' 



The series of Taylor is, • 



(4.) f{^-\-h)=y+-^ + ^2"+^ 2:3+"^ 2:3:4+^^- 



The binomial series is, 



n — 1 n — rl n— r2 



{5.){x+hy=x''-\-nx^~^h-\-n.—^x''~'^h^-}-n.—^.—^x"~^h^r\.hc. 



If in the series of Taylor we make [j']^=[E], and x'=l, and in 

 the series of Maclaurin, if we make z and z' each, equal to 1, they 

 will coincide with the preceding series. These series indicate, what 

 share each of the orders of fluxions has in forming the increment, 

 and disclose the relation of the several orders of fluxions to the flu- 

 ent, included in the following properties. 1. When x-f-x' represents 



