New Mechanical Quadratures. 351 



of terms of Euler's series. This amounts to a choice between 

 formulae (1) to (6) or others similarly derived. Thereafter 

 the only question is how small a value of h must be adopted 

 to give the required accuracy. Any quadrature by series 

 assumes a fictitious curve more or less nearly coinciding with 

 a real one. For the quadratures under discussion the number 

 of derivatives eliminated determines the order of the contact 

 of the two curves at the extremities of the arc to be inte- 

 grated, and also a minimum number of common points on 

 the two curves. By division of h the number of common 

 points is increased in simple proportion to the number by 

 which h may be divided, but the order of contact at the 

 extremities is 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 in- 

 significant errors. For this purpose Bertrand selected 



■ 1 Iog (l+.tW.i ? = g ! 2 = . 2721 c ) 826 . 



1 + x 8 6 



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

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

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

 energy curve of the spectrum. 



I have taken the same example using 8-place natural 

 logarithms and an 8- figure computing machine, and get the 

 following table of results in which I stands for the computed 

 integral, 



i 



f 



J Formula. 



n. 



I= 7r log2 + 

 8 ° 



1 



2 



+0-00181 206 



2 



4 



-0 00002 3G3 



3 



6 



-0 00000 545 



4 



8 



-0-00000 154 



5 



10 



-0 00C00 0G0 



6 



12 



+0-00000 000 



7 



12 



+0-00000 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 



