ON QUADKATURES AND INTERPOLATION. 



355 



not simply to 1 or ± -^, unless the formula be first duly transformed. 

 Proceeding in this way the following formulse are obtained.* 



Expression for the Area f 



a 

 a + 



a + 

 a + 



M 



6 , 



-s^ + ro'i 



-o ^ + on"" 



, 8 , , 86 



a + 



20 



— c + 

 45 ^ 



} 



^ 41 , 

 + 840^ 



} 



25 , , 175 , 3445 , , 4045 , 16067 

 6 ^ 36 ^ 1512 ^ 9072 

 ".58 , , 1833 



, ., , 103 , 158 

 a +66 + ~c + 



_989_ 



28370 ^ 



404^ 16067 ."I 



9072 ^ "^ 598752 -^ J 



B33 , 4813 . 



^ + ll550^ 



700 



1364651 1 

 + 6306300 ^ J 



A similar set of formulae may be obtained, symmetrical to an interval, 

 and in terms of A', B' &c. ; but the coefficients, as well as the form of 

 the series, are more complicated, and the accuracy somewhat less, on the 

 parabolic hypothesis, than the corresponding rule of the other series. 

 The first rule in this set is the common polygonal rule ; the next is got by 



3 3 



integrating from ~ -^ to + ~, and is 



area = 



= 3(A'+iB') 



which is equivalent to Cotes's rule of four ordinates. 



These rules may also be obtained by the direct substitution of sym- 

 metrical diflferences for the ordinates in Cotes's rules, with which they 

 are of course identical, except in form. 



The application of symmetrical difierences to quadratures may also 

 be made to depend upon the formula 



J''dx^^= {log(l + A)}"" 



* See Stirling, Metli. Biff. p. 148. In the formulje for 9, 11, and 13 ordinates, 

 Stirling simplifies the numerical coefficient of the final term, just as he has done in 

 the table of corrections for Cotes's rules, apparently for easier use. The practice is 

 not a satisfactory one, as it prevents verification, and saves but little work. 



t In the above formula3 the interval between the ordinates is taken as unity. If 

 the whole base is taken as unity, the area is given by omitting the numerical factor 



outside the l K The table has been independently computed, and compared with 



Stirling's table. 



A A 3 



