99 OP PRESSURE AND EQUILIBRIUM. 



and substituting // for x — a, and ^% for u — m, A'mz:— . 



Scholium. This proposition, which was invented 

 by Newton, may be applied with great convenience 

 to some cases of interpolations, the constant diffe- 

 rence of time being A, and the variations of any other 

 quantities depending en it being Au and A^w. For 



this purpose, if we make the fraction -—-zim, the theo- 



1 — ^ A a 1 — »* 2 — m 

 rem will become A'MzzmAw — m. — ^r — iX^u + m.—— — — 



A^M — . ..; and these three terms will be abundantly suffi- 

 cient for almost all cases that can occur in practice. 



247. Lemma E. Supposing the quantity 

 X to vary gradually and uniformly, and h to 

 be any finite difference of x., the corres- 

 ponding finite difference of another quantity w, 



depending on it, will be az/'=A. jj +j^ . -5^^ + 



Y2Z'ir^ +...,!/' being the initial value of the 

 quantity u. 



If we suppose the constant finite difference h of the 



preceding proposition to become evanescent, we shall have 



Aw dw ,.^^ A2m d^M , . . .,, , 



■—=-- (46), -y-=:-jrj-, and the equation will become 



/Muzzh! T- + -TK^^u-^ . . ., since h'—h, h'—^h may be qon- 

 da: 1.2 



sidered as simply equal to h\ when h vanishes : and we 



