(subdivided, and the quadrature is applied to each subdivision until convergence is obtained., 

 The series (of integrals) is terminated when the ratio of the absolute value of any one integral 

 to the absolute value of the sum of terms up to that integral is sufficiently small. The first 

 term in the series involves the calculation of a principal value integral but since an even 

 order (10 point) quadrature is used in which the abscissae are symmetric about (but do not 

 include) K , the singularity there can be ignored. 4 



SUPERCRITICAL SPEED (F > 1) 



The significant differences when F > 1 are that $ = cos - 1 ( — J , and there is a log- 

 arithmic singularity at Q = d Q in the integral /j . Because of these the Tchebysheff quadrature 

 is not applicable and the 10-point Gauss-Legendre quadrature is used. 



The integral / 2 is approximated by the same method described previously, namely, sub- 

 dividing the range (d Q , n/2) of integration and applying the quadrature over each sub- 

 division, the number of such subdivisions being whatever is necessary to obtain convergence. 

 The integral I ^ is written as the sum 



where 



/ n = \ d6 \ dK, and 



o o 



77/2 °o 



/ 12 = f d6 I dK 



In both I 11 and / 12 the 6 integral is approximated by using the Gauss-Legendre quadrature in 

 combination with the 10-point Gaussian quadrature for integrals with a logarithmic singularity. 5 

 The integral is approximated in the same way as / 2 , except that the Gauss-Legendre quadra- 

 ture is used over all the subintervals except the one including & for which the logarithm 

 quadrature is used. 



The K integral of / 12 is calculated exactly as in the subcritical case. However, for 

 / X1 it can be readily seen that there is no non-zero root of the transcendental equation and the 

 integrand is well behaved. For this case the integral is written in the same infinite series 

 as before and is approximated exactly the way as for subcritical speed with the exception 

 that K Q = 0. 



By the requirement that sufficiently close estimates or of convergence be obtained, a 

 relative error of less than 1 percent is typically implied. 



15 



