222 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



yo> yi> y*> an d finding the area between the /-axis and this curve and bounded 

 by the ordinates / and fe. The equation of the curve is 



1. y = a Q + 0i (* - t ) + a z (t - to) 2 , 



where the coefficients a 0j a\, and #2 are determined by the conditions that y 

 shall equal yo, y\, and y 2 at / equal to / , h and fe respectively; or 



{ yo = 0o, 



2. \ y\ = UQ + 0i (/i - /o) + 02 (*i - /o) 2 , 



[ >2 = 00 + 01 (fe - to) + 02 (fe - /o) 2 . 



It follows from these equations and /2 h = h t Q = h that 



I 



02 = ~j~2 (y ~ 2 3' 1 ~f~ y*) 



//* 

 ydt is approximately 

 , 



/ = / 0o + a\(t /o) + 02^ - t ) 2 \ dt = 2h\a + aih + - a^h 2 , 



Jh L J L 3 J 



which becomes as a consequence of (3) 



h , 



10.12 The value of the integral over the next two intervals, or from tz to / 4 , 

 can be computed in the same way. If n is even, the approximate value of the 

 integral from t Q to t n is therefore 



FI = - [y + 43>i + 2^2 + 4^3 + 2^4 + + 4y-i + yJ- 



o 



This formula, which is due to Simpson, gives results which are usually remarkably 

 accurate considering the simplicity of the arithmetical operations. 



10.13 If a curve of the third degree had been passed through the four points 

 yo, y\j yz, and y 3 , the integral corresponding to (4), but over the first three 

 intervals, would have been found to be 



I = DVo + 33>i + 3^2 + yal 



10.20 Digression on Difference Functions. For later work it will be necessary 

 to have some properties of the successive differences of the values of a function 

 for equally spaced values of its argument. 



As before, let y be the value of /(/) for t = h Then let 



