SIMPSON'S RULE 251 



Let three strips be taken together, and let the axes of reference 

 be so chosen that the axis of y comes midway between the ordi- 

 nates t/ a and t/ 3 (Fig. 77). 



The curve must pass through the four points A, B, C, and D, 

 and will be of the form y a + bx + cx z + dx 3 where a, b, c, and d 

 are constants. 



* 2i 4 O 



x = - V = Va Va = + ~bh + -ch 2 + -dh 3 (3) 



2 248 



3/i 3 9 27 



and these equations can be solved for the constants. 



Area of the three strips = y dx 



~~T 

 ft 



(a + bx + ex 2 + dx 3 ) dx 



J ~Qh 



T 



M 



r i i ii 2 



= \ax + -bx 2 + -ex 3 + -dx 4 



* A 4 J-3A 



A! = Sah + 







Thus to express A x in terms of the ordinates, it is only necessary 

 to find the constants a and c. 



A 



Adding (1) and (4) y^ + t/ 4 = 2a + -ch 2 

 Adding (2) and (3) y z + y 3 = 2a + 



Then yi+y*-y z - 



1 



Also 9(y 2 + j/ 3 ) - (y l + t/ 4 ) = 16a 



1 , 



