372 proceedings: philosophical society 



mulas are known for evaluating definite integrals of functions whose 

 primitives are unknown. The most familiar formulas are those devel- 

 oped by Cotes, Lagrange, Euler, and Gauss. In each of these the 

 problem is reduced to finding a certain area under a curve representing 

 the function in rectangular coordinates. This area is given in terms 

 of the interval of integration and several ordinates (values of the func- 

 tion) within that interval. Thus, to a certain degree of approximation, 



f{x)dx = ih-a)^[krf{x,)]. (1) 



j=l,2, 3, ... 71. 



The numbers /(xi) and /^i are ordinates and corresponding " weighting" 

 coefficients in the interval (a. . . .6). 



In the formulas of Cotes and Euler the ordinates are equidistant. 

 In those of Lagrange the ordinates may be taken at random. All three 

 types have equal accuracy of a certain kind, namely, a formula using 

 n ordinates gives the exact value of definite integrals of a polynomial 

 function of degree n — 1 (or degree n if n is odd). In the formulas of 

 Gauss, however, the ordinates must be taken at certain definite points, 

 and in virtue of this the maximum accuracy is obtained. A formula 

 of n ordinates is exact for the integral of a function of degree 2 n — 1, 



It can be easily shown that the use of any type of ordinate formula 

 is valid only in the case of analytic functions (i.e., such as can be de- 

 veloped in a convergent power series) . From the properties common to 

 all analytic functions, it may be shown that the points Xi and the co- 

 efficients k of the corresponding ordinates subsist in a unique functional 

 relation. Thus by a simple transformation, we may put 



J»6 p + 1 



/ {x) dx= \ if (t) dt. (2) 



a J-\ 



Then, it can be shown that 



J^ <p{t)dt=2-^k,[^{-r)-^^{+r)] (3) 



^ = 1, 2, 3, . . . n, 



O <n<r.<r, . . . 1\<\, 



in which the numbers ki and /-j are each arbitrary but related as follows: 



1 



' ' 2(2r/+l) ^^ 



-7 = 0. 1,2,3 



This relation determines an infinite number of types of approximate 

 integration formulas, combining maximum accuracy with maximum 

 flexibility. For k^ and r, may have arbitrary values, consistent with 

 their fundamental relation, and subject only to the condition that the 



