G. F. Becker — New Mechanical Quadratures. 



123 



Comparison with (1) shows that it is somewhat less accurate 

 than Simpson's rule. Newton's rule is derivable also from the 

 integration of his interpolation formula, and if 3A is taken as 

 unity it coincides with Cotes's rule for n — Z. The lowest odd 

 number with two divisors is 9, so that with n=9 two deriva- 

 tives could be eliminated, but such a formula would be of 

 small value. 



It is noteworthy that the simpler rules for quadrature are 

 derivable from any one of the three fundamental interpolation 

 formula. 



Other formulas coidd be obtained by eliminating fewer coef- 

 ficients than the divisibility of n permits. In general that 

 would be a waste of opportunities, but two examples are worth 

 noting. With n=6, I find from T, C 2 , and C 3 



/ 



OCq 



OCr 



15T + 3C. + 2C 3 A 6 v T 



ydx — ? 6 A. 



* 20 5,040 



+ 



which is Weddle's rule with a corrective term. Under normal 

 circumstances it is considerably less accurate than (3), as can 

 easily be shown by applying each of the equations to the same 

 portion of the exponential curve. 



Curiously compact and accurate is a formula derived from 

 T, C 2 , C 3 , and C 4 , in which the coefficient of C 2 turns out to be 

 zero. Of course n must be 12 or a multiple thereof. It may 

 be written 



cc u 



J yd* 



4T + 



8C„ - C. 



12 A. 



A 7 



25,200 



(*)• 



Here the denominator of the corrective term is remarkably large, 

 or the remainder very small, so that (7) may approach (6) 

 in accuracy. Its simplicity makes it convenient for laboratory 

 use. Economy of labor depends to some extent on the arrange- 

 ment of computation, and for that reason I give in a footnote* 

 (p. 124) the details of the quadrature by (7) of a portion of the 

 ascending exponential. It will enable the reader to perceive 

 that no advantage is obtained by stating the formulae in terms 

 of the ordinates instead of the polygons, even when the divi- 

 sion of the area to be integrated is limited to the minimum 

 value of n. 



It is needless to say that the integrals (1) to (6) without the 

 corrective terms are rigorous for finite series with n + 1 con- 

 stants whose highest terms contains x n . In any other case two 

 distinct means exist for reducing the error of the result below 



