OF PRESSURE AND EQUILIBRIUM. 91 



Again AmzzMj— mi= 



^2— 2mi+m 



A'mA=A(A2m)=Am2--2^"i +^" 



-2^2+22/1 



Here the operation of the characteristic A at each step 

 doubles the number of terms to be added together, the 

 coefficients being always formed, as in involution, by the 

 addition of two contiguous ones of the former step : con- 

 sequently the same law prevails as in the dinomial theorem. 



246. Lemma D. If a constant finite dif- 

 ference of X be called A, and any other diffe- 

 rence h\ the difference of u^ corresponding to 

 A, being Aw, that which corresponds to K will be 



Since m^= m + wAm + ti.— ZL A 2 u-\- ... (245), if we suppose 



a+nhzzx, X representing an absciss of which u is an ordi- 

 nate, and a the initial value; or, in other words, m being a 

 function of a;, and the difference Am corresponding to the 



difference hzzAx, substituting for n its value — r— > ^^ 

 shall have u^ =u + _.A» +t^.(-^-l)_j3 + . . . ; 



