Xew Mechanical Quadratures. 347 



so that the (/c + l)st difference is negligible, then 



k (k) (k) 

 hf(x) = A 



where A denotes finite difference. In the formulae v is 

 employed to indicate the difference of the derivatives at the 

 limits of the area to be integrated, x Q and x n . Using a 

 corresponding notation for the finite differences and assuming 

 that the (/fc + l)st difference is inconsiderable 



(k) (k) 



7tV=A n -A , 



and this substitution may be made in the corrective terms of 

 the formulae *. 



Not all of these equations are wholly new. The first term 

 of (1) is only Simpson's rule in a new notation and if n is 

 limited to 2 it is also identical with Cotes's rule for n = 2. 

 Omitting the derivatives, equations (2) and (3) also coincide 

 with Cotes's rules for ?i = 4 and a— 6, but it' in these equations 

 n is taken at any multiple of 4 and 6 numbers quite distinct 

 from Cotes's result. All of the equations can be expressed in 

 the same foi-m as Cotes's, but this mode of statement seems 

 undesirable because it masks the vital i'act that a reduction 

 of the value of h increases the accuracy of the result. Now 

 no one would think of getting a considerable quadrature by 

 Simpson's rule with the minimum value of n = 2, because 

 this rule with n = 10 gives a result the error of which 

 approaches a 625th of that incurred by taking n at 2, while 

 if in (6) n is taken at 24 instead of 12 the error is reduced 

 approximately to 1/4096 of its maximum value. 



So far as I know, equations (4), (5), and (6) are new, and 

 Cotes's numbers for n = 8 and 10 do not fit into the system 

 of quadratures here discussed. 



The derivative term in each of the six equations may 

 exceed the value of the remainder. If the difference of 

 derivatives in this term is denoted by v r , this is to be regarded 



0;+i) 

 as the definite integral of f(x) which, like any other function 

 of real variables to be integrated, must preserve the same 

 sign between the limits of integration. If the sign does not 

 change and if also (as Poisson and Jacobi showed) the 

 (r-f-l)st derivative does not pass through a maximum 

 between the limits, then the final term of the equations 

 exceeds in absolute value the remainder of the series. In 



* A discussion of the relations subsisting between derivatives and 

 finite differences may be found in Smithsonian Math. Tables, 190a, 

 page xxxvi, or elsewhere. 



