G. F. Becker — New Mechanical Quadratures. 125 



not affected by this process. The remainder, on the other hand, 

 is inversely proportional to a power of the number by which h 

 is divided, a power greater by one than the order of the 

 retained derivative. 



In order to test the accuracy of formulae for mechanical 

 quadrature it is clearly necessary to take a difficult example, 

 for otherwise all reasonably good formulae would show insig- 

 nificant errors. For this purpose Bertrand selected 



/ 



lQ g f 1 + X) dx = ^log 2 = 0-27219 826. 

 I + X 8 ° 



The curve in this case rises sharply from the origin, passes 

 through a maximum at x = 0*7825. . . and then approaches the 

 &--axis asymptotically. In general form it resembles the 

 energy curve of the spectrum. 



1 have taken the same example using S-place natural loga- 

 rithms and an 8-iigure computing machine, and get the fol- 

 lowing table of results, in which I stands for the computed 

 integral : 



Formula 



n 



1= — log 2 + 



8 



1 



2 



+ 0-00181 206 



2 



4 



— 0-00002 363 



3 



6 



— 0-00000 545 



4 



8 



— 0-00000 154 



5 



10 



— 0-00000 060 



6 



12 



± o-ooooo 000 



7 



12 



+ o-ooooo 001 



The results for (6) and (7) are very satisfactory, while those for 

 the earlier formulae could be greatly improved by taking n at 

 a multiple of its minimum value. According to Bertrand. 

 Gauss's formula for %=4 gives a result which is too small by 3 

 units in the seventh place, and thus tested it is intermediate in 

 accuracy between (5) and (6).* 



* Eather curiously WedcUVs rule applied to Bertrand's problem gives 

 somewhat better results than (3). Furthermore, as this rule is deduced 

 from Newton's interpolation formula, it appears to err only by a small frac- 

 tion of the sixth difference, when the seventh difference is negligible. As 

 here deduced from Euler's equation, the error should include fifth differ- 

 ences. "While these facts are not incompatible, the relations seem to need 

 confirmation, andlhave integrated e x dx from x=—l to as=2"6, taking 7i = 0'6 

 and using values of e* with 7 significant figures. By a separate computation 

 I find the true value of the integral to be 13'09585 85938. Weddle's rule 

 gives a value which is too great by 00064. while the value of the corrective 

 term given in this paper for his rule is -00073. or about 9/8 of the real 

 error. Formula (3) gives a value - 00016 too great or 1 4 of the error of 

 Weddle-'s rule and 2/3 as great as the value of the derivative term in (3). 



