336 REPORT— 1880. 



Stopping at m = 1 , 2 m + 1 = 3 



^1 + /'s = 3, jui + 9^3 = 9 



X 



3 9 , 



..^3 = ^,^1 = ^, and 



3 



udx=-- («,_3 + Sit.i + 3 zfi + u{) 



— 3 Ti 



It is to be observed that tbe interval in the /j, formula is 2, and not- 

 unity. 



The actual coefiScients for quadrature by means of equidistant; 

 ordinates, when the interval i^ taken as unity, are* 



2 ordinates or 

 1 interval 



3 ordinates or 

 2 intervals 



s or "1 

 al / 



} 



4 ordinates oi' > 



3 intervals J 



5 ordinates or 



4 intervals 



6 ordinates or 



5 intervals 



7 ordinates or 



6 intervals 



8 ordinates or 



7 intervals 



9 ordinates or 



8 intervals 



10 ordinates or 



9 intervals 



8 



45 



J_ 



288 



(l+.L). . 

 (1 + 4 + 1) 



The trapezoidal rule 



The parabolic rule, or 

 Simpson's first rule 



(1 + 3+3 + 1) Simpson's second rule 



(7 + 32 + 12 + 32+7) 



(19 + 75 + 50 + 50 + 75 + 19) 



,4k (41 + 216 + 27 + 272 + 27 + 216 + 41) 



140 



7 (751 + 3577 + 1323 + 2989 + 2989 + 1323 

 17280 +3577 + 751) 



4 (989 + 5888 - 928 + 10496 - 4540 + 10496 

 14175 - 928 + 5888 + 989) 



9 (2857 + 15741 + 1080 + 19344 + 5778 + 5778 

 89600 + 19344 + 1080 + 15741 + 2857) 



11 ordinates or "1 10 (16067 + 106300 - 48525 + 272400 - 260550 

 10 intervals J 598752 + 427368 - 260550 + 272400 - 48525 



+ 106300 + 16067) 



In the. foregoing the numerical coefficients only are given, and the 

 ordinates have to be inserted. Thus, the ordinates being a, b, c, d, e, the 

 rule for five ordinates or four intervals is 



^^ (7a + 32&+ 12c + 32(7 + 7e) x interval. 

 45 



To the above should be added Weddle's approximate rule for 7 

 ordinates, or 6 intervals, namely : 



3 



10 



(1+5 + 1+6 + 1 + 5 + 1). 



* These are all taken from Cotes, Harnwnia Mensurarum, by Robert Smith, 

 Cambridge, 1722, De Methodo Bifferentiali, p. 33. The principle of these rules seems 

 to have been known to Newton ; see his MethoduB Differentialis already quoted. 



