of Definite Integrals. 435 



which for any positive and integer value of n is always O; and 

 therefore the given series is simply equivalent to 



which is evidently equal to the definite integral of ^(j). 



Most of tliese theorems may, without considerably altering 

 their forms, be applied to the definite integrals in finite diifer- 

 ences, but it would too much extend this paper to insert them 

 with their demonstrations: the following, for instance, is analogous 

 to Theorem III. D being used to denote a definite integral in 

 finite difl^erences, a and a + mh the limiting values of x, h being 

 the increment of x or a, then we shall have 



Theorem V. 



^ rv y J 2 3 ■ 2 



W2-2 ^.(pAa) m + 2 A.d),(a) . 

 + -^.— ^ 3~- 2 +*'■ 



where (p„{a) denotes the error made by taking («-l) terms of 

 the series for the true definite integral. 



From this theorem and any others of the same nature, in 

 which each term de{»ends on the error made by taking a certain 

 number of the preceding terms for the whole, many remarkable 

 theorems with respect to the reproduction of functions, even when 

 not continuous, may be deduced : for example, 



* If there be n rows of quantities, the first row entirely arbitrary, 

 any even row, as the 2m'\ formed so that the vertical difl^erence 



• In series formed after this manner, it is a curious property tliat the difiereiice of 

 terms in n"" row, which ; 

 terms in the row preceding 



wo terms in n'" row, which are —^^ places distant = ^^-^ x difference of two consecutive 



