338 



REPORT — 1880. 



putation by the rule for 2m — 1, or 2 «i ordinates, over the actual value of 



the ini 



integral / : 



,2 m 



dx. 



Number of 

 ordinates 



2 

 3 

 4 

 5 

 6 



Number of 

 intervals 



Excess 



_1 

 6 

 1_ 



120 



1 

 270 



1 



2688 



11 

 52500 



Excess 



38880 



167 



10588410 



37 

 17301504 



865 



631351908 



260927 



136500000000 



This is a fair indication of the error to be expected in treating a con- 

 vergent form by these rules. It is no criterion where the curve approaches 

 parallelism to the ordinate. 



It must be remembered also that the higher rules use more ordinates, 

 and therefore ought to give more accuracy. As regards relative accuracy, 

 the proper test is so to use the rules as to cut up the function or curved 

 area into the same number of intervals, for vehich purpose it is necessary 

 to use the least common multiple of the order of (or number of intervals 

 in) the rules. Thus, v?hat has hitherto been considered, in the case of 

 two and three intervals, is the comparison of 



vrith 



+ 1 



|{M(0) +3^(1-) 4- 3^(1-) 

 the proposed comparison is between 



and |( 

 with corresponding ordinates, namely 



•«>(i) } 



) 



1+4+2+4+2+4+1 



1 + 3 + 3 + 2+3 + 3 + 1 



) 



Kl) 



In this way, using 21 ordinates, or 20 intervals, 



1+10 



x^dx gives 



/ 



- 10 



Aecarate value by integration 

 By rule of 5 ordinates 

 By rule of 6 ordinates 

 Eatio of errors 128 : 275 



2,857,142f 

 2,857,173i 

 2,857,208^ 



Errors 

 + 



+ 



30ii 



66^^ 



