New Mechanical Quadratures. 319 



Odd values o£ n lend themselves less readily than even 

 ones to the elimination of derivatives from Euler's series 

 because of their limited divisibility. If /i = 3 the quadrature 

 may be written 



.f 



, 9d-C3 o ; Wv' 



ydx — 



8 210 



and the first term of this formula was given by Newton *. 



Comparison with (1) shows that it is somewhat less accu- 

 rate 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 Gotes's rule for n = 3. 

 The lowest odd number with two divisors is 9, so that with 

 n = 9 two derivatives.could be eliminated, bat 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 interpola- 

 tion formulae. 



Other formulae could be obtained by eliminating fewer 

 coefficients than the divisibility of n permits. In general that 

 would be a waste of opportunities, but two examples are 

 worth noting. With ?i = 6 I find from T, 3 , and C 3 



r 



15T + 3C 2 + 2C 8 CI ftV 



ydx = • : 2 ~ A - 6 h-—^ + 



20 5u4U 



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 



J^-sO 



, 8C3-CA . /^- 

 +___j_i 27i + . . (7) 



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 labour depends to some extent 

 on the arrangement of computation, and for that reason I 

 * Opuscula, Methodus differentiates, prop, vi, Scholium. 



