90 OF PRESSURE AND EQUILIBRIUM. 



other cases : but for the present purpose, that of showing 

 the analogy to the laws of differences, the integral powers 

 are suflScient. See 2/8. 



245. Lemma C. If Az/, a'^w .. .be the suc- 

 cessive finite differences of the quantities 

 u, u ,u , . , ^ we shall have u —u-\-nlu+n* 



12 n 



n—l , , ^ n—l 



In the first place 



Au^z:zu,-u, ...._.' .. A^u, = A^u,-^A^u, 



'2 — "'s --2 A^M^^AWg — AMg 



'^''^-""^"''^ A^u-A^u,-^A^u 



Hence, 



Mj^rzM -\-Au 



m^^Mj+AMj =z m 4- Am + A(m + Am) =:m + Am 



+ Am + A2m= 



tt3=M2+AM2=:M + 2AM + A2M ( + AM2) u + 2Au-\-A^u 

 + A{uA Au + Q>A^u-\-A^u= 



u + SAu + dA^u + A^u 

 Now the steps of this operation are just the same as if 

 we multiplied each time by 1+A, though the symbol A" is 

 not exactly a power of A : but we may always ; make 

 AwjZiAM + A^MwhenM^ is =:M + AM,which is in itself suffici- 

 ently evident, and is also shownbythe equation A^u=Auj^ — 

 Am whence Am^izAm + A^m. The process is thus obviously 

 similar to that of involution, and the law of the coefficients 

 must be the same (244.) This method of reasoning, ap- 

 plied to the eye only, has been much extended by La- 

 grange, Arbogast, and others. 



