G. F. Becker — New Mechanical Quadratures. 117 



Art. XII. — Some Nexo Mechanical Quadratures ; by George 



F. Becker. 



Mechanical quadratures are in some circumstances unavoid- 

 able, but they are usually shunned as clumsy and troublesome. 

 Were the formulae neater and their applicability better defined, 

 they might be of great use in experimental physics and might 

 compete with other integrations by series in the computation 

 of functions. This paper is intended as a contribution to that 

 end. 



There are three distinct systems of mechanical quadrature 

 each depending upon the integration of a general formula for 

 interpolation. By integrating Newton's interpolation for- 

 mula Simpson's rule, Weddle's rule, and some others can be 

 found. These depend upon the assumptions that the incre- 

 ments of the abscissa are constant and the differences above a 

 certain order are negligible. It is said that if the attempt is 

 made to include in quadratures of this species differences above 

 the sixth, the formulas become unmanageable, but this I have 

 not tested. It is possible to base quadratures on Stirling's and 

 Bessel's interpolation formulae, but as these are not appli- 

 cable at the beginning or the end of a series of values their 

 usefulness is limited, and since these modes of interpolation 

 are founded on Newton's, the quadratures are not essentially 

 different. 



Gauss used as the basis of his method of quadratures La- 

 grange's interpolation formula. When after integration the 

 increment of the abscissa is assumed to be constant, Cotes's 

 numbers result, but Gauss showed that, by a proper selection 

 of unequal increments of the independent variable, formulae can 

 be deduced whose convergence is very rapid. Unfortunately 

 most of the increments are surds, rendering the application of 

 the method very laborious in spite of its elegance. 



Interpolation by means of Taylor's series has several advan- 

 tages over methods depending upon finite differences and its 

 limitation to continuous functions is rarely of any moment. 

 In dealing with known functions the methods of infinitesimal 

 calculus are habitually employed excepting for interpolation, 

 while finite differences constitute a distinct algorithm. Since 

 Taylor's series is the very foundation of analysis, its application 

 to interpolation is both more consistent and more elegant than 

 that of finite differences, while, if needful, the final results can 

 be expressed in terms of finite differences without the least 

 trouble. Similarly so-called mechanical quadratures founded 

 on Taylor's series in its application to interpolation are quad- 



Am. Jour. Sci. — Fourth Series, Vol. XXXII, No. 188. — August, 1911. 

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