ON gauss's theorem for quadrature, etc. 385 



On Gmiss's Theorem for Quadrature and the approximate Evalua- 

 tion of definite Integrals ivith finite Limits. Bxj Prof. A. K. 

 Forsyth, F.K.S. 



(Ordered by the General Committee to be printed in extenso.) 



1. Most of the rules commonly used for the mechanical quadrature 

 of smooth non-re-entrant curves require the measurement of ordinates 

 at equal distances apart ; the greater the number of ordinates, the closer 

 the approximation. The same remark applies to the approximat 

 evaluation of definite integrals. If, in particular, n ordinates are used 

 the process can fairly be described as taking the ordinate of the curve 

 or the subject of integration, at the value x as equal to 



CTq + Wi* + ... + a„-iX"-\ 



and implicitly determining the n constants a by means of the magnitudes 

 at the n places. 



There is, however, a remarkable theorem due to Gauss whereby the 

 accuracy of the approximation can be nearly doubled. The ordinates are 

 to be measured at places, not equally distant apart, but at selected 

 positions in the range. These selected positions do not depend upon the 

 special curve or upon the form of the subject of integration. For an 

 assigned number of ordinates, the division of any range is exactly similar 

 to that of any other range, and depends only upon this assigned number. 

 So far as is known to me, the rule or rules derived from the theorem • 

 are not applied in ordinary practice ; and my purpose in this note is to 

 recall attention to the utility of these rules, by giving them in a form 

 that requires nothing more than numerical substitution. 



The theorem can be enunciated as follows : — 



The approximate value of the integral I f{x)dx can be made as 



accurate as though 2n— 1 equidistant ordinates are measured, if only n 

 ordinates are measured at the places given by the roots of the equation 



The mathematical statement of the theorem can be simplified by 

 substituting 



■■>i^ = \ {p -^ (f) + h{p-i)y' 



and taking 



f{x) = <l> (y). 



' In its original form, the theorem was stated and proved by Gauss in his memoir 

 Methodus nova .... inveniendi (1814), Ges. Werke, iii., pp. 163-196. A different 

 proof was given by Jacobi, Ges. Werke, vi., pp. 3-11 ; and it is reproduced in 

 Moulton's edition of Boole's Finite Differences (p. 52). Part I. of the second volume 

 of Heine's Kugelfunctionen may be consulted. 



But, as already remarked, the theorem is not applied in ordinary practice, although 

 its theory has received considerable additions. 



