340 APPENDIX. B. 



Corollary. The fluxions, thus obtained, will enable 

 us not only to find any intermediate values of the variable 

 quantity m, but also the areas contained by these values as 

 ordinates, the contents of the corresponding solids, or any 

 other derivative quantities. If it were required, for ex- 

 ample, to determine the magnitude of an area contained 

 between a curve and its absciss from four equidistant ordi- 

 nates, affording us four successive differences, Au, A^m, 

 A^u, and A*u ; we should have to deduce the four succes- 

 sive fluxions from the four first terms of each series, which 

 would afford us, by substituting the values derived from 



the expression A**m=m«'— wMw_i + ..., that is, Amzim — u, 



1 

 Ahizzu — 2tt +M, A^MzzM — 3w +3m — m, and A'^'uznu 



2 1 3 2 1 4 



— 4m -{-Qu — 4m +m, the equations -r— h z= — ^m + 4m 



3 2 1 "-^ 1 



-3m -{-iu -iu , ^7f~zz^u-%^u +y^ - ¥w + 



2 3 4 ax 12 3 



J^M , -j^ h^=:—^u + 9u ~12m +7u — |m , and —^ A* 



4 QX^ 1 2 3 4 OX 



zz u — 4m -\-6u — 4m H-m . Then if we multiply each 



1 2 3 4 



of these expressions by dh, considering h as variable, 



and take the fluent for the whole length 4h, we shall have 



1 . , , .• ^ . . u 16/i 64h 256A 



to multiply the respective coefficients by — -, —^, — ^ 



and ]^^\ or if 4A=/, by 2/, i^h 16/, and ^Ac/, and to 

 5 



divide them by 1, 2, 6, and 24, making the multipliers 21, 



fZ, f/, and j^l; the whole being equal to 



(• 



/ /— 2_5«+8^ _6m +fM 



12 3 4 



12 3 4 



