A Theory of Fluxions. 347 



%xx\ hence A=l ; also =a; ,2 ,hence B=2. Again ' 



B 6 A 



firr' 2 fir -3 



+'ri_+ D _=3o; 2 a;- 4-3^-2+0; •*. Let the corresponding 

 terms be taken, and x x =3x 2 x-, hence A=l ; also 



B 



6a;* 3 

 =3a;x ,a , hence B=2 ; likewise _-=a;- 3 , hence C=6. Put- 



tingx 3 =E, it will be, ' — -f- = the corresponding 



increment of a; 3 . By taking higher powers, we may extend 

 the series to any proposed length. Taking for instance a; 4 , 



and proceeding as before, we obtain * — -f- — + — = 

 v S 2 6 24 



the corresponding increment of a; 4 , and so on for other 

 powers. 



If the power of a; is supposed to be infinite, then the fore- 

 going series also becomes infinite ; and if the digits which 

 produce the several multiples are taken, we have, 

 E-+E" E;- + E-^ JS-^ & ^ . n . nf _ the c( ^ 



2 2.3 2.3.4 2.3.4.5 

 responding increment. 



Whence the law of continuation in the series is manifest. 

 This series may be derived from the celebrated theorem, pub- 

 lished by Taylor in his Methodus Incrementorum, by making 

 z=z\ (See Maclaurin's Flux., Vol. 2, Sec. 751,) and it hence 

 appears, what share the fluxion of each order contributes to- 

 wards producing the increment of any proposed power. 



Fig. 9. 



%— » Let ABC be a half square, AC=BC 

 ='*■ CE =BF =ar. The first fluxion is 

 the parallelogram BFEC=a;a;•, the se- 

 cond fluxion is the small square GDFB 

 =x 2 . When %• is invariable, x' a is a 

 constant quantity ; hence the third flux- 

 ion is equal to 0, and the series of flux- 

 ions here ends. By the figure it is man- 

 Jl~ c """e ifest, that the increment BDEC is com- 



posed of the first fluxion BFEC, and half the second fluxion 

 DBF. 



