ON QUADRATURES AND INTERPOLATION. 



339 



/20 



X gives 



Accurate value by integration . . 182,857,142|^ Errors 



By rule of 5 ordinates . . . 182,857,173^ + 



By rule of 6 ordinates . . . 182,867,208^ + 



which accords -with the former result. 



In the same way, using 7 ordinates or 6 intervals; we find that 



>+3 



x'^dx gives 



301^ 

 66i^ 



/: 



Accurate value by integration 

 By rule of 3 ordinates 

 By rule of 4 ordinates 

 Ratio of error 4 : 9 



97-2 



98 



99 



Errors 

 + 0-8 

 + 1-8 



Again^^] 



x^dx gives 



1555-2 



1556 



1557 



Errors 

 + 0-8 



+ 1-8 



arise from 



Accurate value by integration . . 



By rule of 3 ordinates .... 



By rule of 4 ordinates .... 



which accords with the former result. These coincidences 

 the change of origin not affecting the definite integration. 



In this particular case the errors may be shown symbolically 



operating at once upon ( 1 + -^ ) ' from a; = to re = 6 by 



(1) Simpson's first rule (three ordinates) 



(2) Simpson's second rule (four ordinates) 



(3) The rule of 7 ordinates 



The results are, in ascending powers of A 



by 



(1) 



(2) 



6 + 18 + 27 + 24 + 12 i- + 



3i+i 

 "^3 + 3 



(3) . . , 



and the errors are 



for (1) 

 for (2) 



6 + 18 + 27 + 24 + 12-1 + 



o 



6 + 18 + 27 + 24 + 12^1 + 



+ JLa* +— A-5 +— A6 

 30 ^30 420 



8 



3A 

 10 



-I 



+ 



41 



140 



40 ^40 



A5 + 



23 



280 



AG 



in 



\ 



Neglecting the last term, it appears that the ratio of error is as 4 

 favour of Simpson's first rule as against the second. 



It is worth while to continue this comparison backwards. For thus 

 it is not necessary to have recourse to arithmetic. Taking a parabola with 

 axis parallel to the ordinates, it is easily seen that the rectangle between 

 the middle ordinate and the base is a better approximation than the 

 trapezium consisting of the chord, the base, and the extreme ordinates, 

 and that the ratio of the errors is + 1 : — 2, the errors being of opposite 

 sign. 



So far as the first six cases go, therefore, it appears that a rule with 



z2 



