226 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



It follows from the last equations of these four sets of equations that 



60 = y n +i - Ai>' n+ i, 



11 A -* A 



o 



- A 2 )>n+l, 





Therefore the integral (3) becomes 



~ . . . . I. 



The coefficients of the higher order terms A 4 ;y n+ i and As;y n +i are - and 



respectively. 

 4 y 



10.31 Obviously, if it were desired, the integral from / n _ 2 to t n -i, or over any 

 other part of this interval, could be computed by the same methods. For example, 

 the integral from / n _i to / is 



24 



NUMERICAL ILLUSTRATIONS 



= kly^i 



\_ 



10.32 Consider first the application of Simpson's method. Suppose it is required 

 to find 



^55 r ~\55 



I = I sin / dt = \ cos / = 0.3327. 



J2 5 J 25 



On applying 10.12 with the numbers taken from Table I, it is found that 



,-0 



7i = [.4226 + 2.0000 + 1.1472 + 2.5712 + 1.4142 + 3.0640 + .8191], 



o 



which becomes, on reducing 5 to radians, 



/i = 0.3327, 

 agreeing to four places with the correct result. 



10.33 On applying 10.11 (4) and omitting alternate entries in Table II, it is 

 found that 



/ = lsmtdt= [.4226 + 2.2944 + .7071] = 0.1992, 



t/2 5 3 



which is also correct to four places. These formulas could hardly be surpassed 

 in ease and convenience of application. 



