43 



The velocity U(t) will be determined at the 16 points along the body 

 whose abscissae are t. = §,, the Gaussian values for the l6-point quadrature 

 rule, Table 1. Since the body is symmetrical fore and aft, it is necessary to 

 determine the velocity at only half of these points. Values of y(x), cos y(x) 

 and k (t) for these points are given in Table 6. 



In order to apply the Gauss l6-ordinate rule it is necessary to eval- 

 uate the integrands in [109] an( * [110] at the l6 Gaussian abscissae x, = £. 

 for each of the 8 values of t, . Thus, there are l6 x 8 = 128 values of k(x, t) 

 and of k 1 (x, t) to be determined. The matrices K.. = R.k(x,, t.) and 

 K 1 .. = R.k'(x., t, ) where the R.'s are the Gauss weighting factors, are given 

 in Tables 7 and 8, and applied to evaluate E (t) from [109]. E , E , and E 

 are then obtained from [110]. U (t) is then given by [102] and then p/q by 



2 5 



[69], in the form p/q = 1 - U . The arrangement of the calculations and the 

 results are given in Table 9- The graph of p/q is included in Figure 5- 



TABLE 6 



Values of y, cos y , and k (x) for Application of 

 Gauss l6-Point Quadrature Formula 



X 



y(x) 



y'(x) 



y(x) 



cos y(x) 



k x (x) 



-0.9894009 



0.0408548 



1 .8965483 



1 .0856 



0.4664 



0.096382 



9445750 



.0903198 



0.7464764 



0.6412 



.8014 



.093389 



.8656312 



.1324422 



.3917981 



■ 3734 



• 9311 



.088359 



.7554044 



.1642411 



.2099651 



.2070 



• 9787 



.081862 



.6178762 



.1848527 



.1020867 



.1017 



.9948 



.074689 



.4580168 



.1955501 



.0393076 



•03932 



O.9992 



.067885 



.2816036 



.1993706 



.0089607 



.008961 



1 .0000 



.062506 



-0.0950125 



0.1999919 



0.0003431 



0.0003431 



1 .0000 



0.059509 



