120 G. F. Becker- -New Meohanioal Quadratures. 



into 60 parts, eleven coefficients could be eliminated and the 

 integral would be expressed in terms of T and eleven polygons 

 of chords only the 23d and higher derivatives or differences 

 appearing in the remaining portion of the series. For certain 

 classes of functions this might be advantageous, but on account 

 of the semi-convergence of Eider's series the desirable limit 

 "will in many instances be lower. 



I have carried out the process only as far asw = 12, which 

 permits of the elimination of all the derivatives below the 

 eleventh. The equations themselves show the . appropriate 

 factors, while the coefficient of the one derivative term retained 

 is the sum of the coefficients in the several Elder series each 

 multiplied by one of these factors. The following six formulas 

 are thus obtained : 



,.. r, 2T + C , AV" 



(l)Jydx = - ir ±-2h. —+.-.. (n>2) 



... 32T4-12C 2 + C, 7iV 



(2) =- _-— *-**.— +.... (n>4) 



= 648T + 8lC a + 112Q n -C a _ AV^ 



v ; 840 5,600 ^ = > 



, _ 2,048T + 704C, + 84C 4 -C e _ 2AV' 1 



1 ' 2,835 ' 4,725 ' 



(6) 



K>8) 



= 35,000T + 14,375C 2 + 528C 6 -7C 10 1QAV" 



' ' 49,896 '12,096 



(n^lO) 

 1,492, 992T— 174,960C 2 + 585,728C 3 

 1,801,800 



104,247C 4 + 2,288C.— C,, 691 A 1 V 



1,801,800 " ' 750,750 + ' 



(n> 12) 



The derivatives in the last or corrective terms of these equa- 

 tions may be expressed in terms of finite differences should the 

 latter be more convenient. The transformation is well known, 

 but its most essential features may be noted here to save a 

 reference. The kt\\ derivative of a function, f (a? (k) ), multiplied 

 by the &th power of the constant increment of x, here denoted 

 by A, is expressible in terms of the kih finite difference and 

 differences higher than the Ztfth. For the purpose in hand New- 

 tonian differences should be employed because they are appli- 

 cable at the beginning and at the end of a series of values. 

 When the derivatives and differences are so large that higher 



