, 2 = J- 



y 2gx 



[y - (B Q + n)] 2 d£ d n . (34) 



These expressions are measures of the "curvilinearity" of the trans- 

 forms x(£,n) and y(£,n) • From equations (9) and (10), it can be 

 shown that 



N 

 a 2 = I a n 2 , (35) 



n=l 



where the contribution to the total curvilinearity variance due to 

 the n tn term is : 



(B 2 + C 2 ) sinh 2nkg 



o = . . . (36) 



n 4nk3 



The value of a n 2 /a 2 is a more meaningful measure of the relative 

 importance of the various terms in the transformation than the raw 

 B R , C coefficients. The curvilinearity variance spectrum given 

 by a n 2 /a 2 as a function of n for the final transform of the five 

 regions is shown in Figure 20. The values of a 2 and o/X for' the 

 five regions are indicated in the figure. Each of the curves indi- 

 cates a general inverse power law trend of about 4th degree. The 

 variability of the spectrum in all but the middle curve is more pro- 

 nounced in the higher harmonics than in the lower. 



3. Limitations. 



Testing of the conformal mapping equations was conducted with 

 coastlines that varied in the degree of smoothness and regions that 

 varied in A . Another version of the coastline from Cape Kennedy 

 to Pamlico Sound is shown in Figure 21. Comparing Figure 5 with this 

 version, it can be seen that there are two small differences in the 

 form of the coastline. Figure 21 shows the narrowing of the Con- 

 tinental Shelf in the area south of Cape Kennedy. Another differ- 

 ence is the slightly sharper vertex at Cape Kennedy, Cape Fear, and 



Cape Lookout. The (general) solution of the mapping equations proved 

 unstable for this situation. It is not known which change in the 

 coastline permitted the convergence of the iterative procedure for 

 Figure 5. 



One test of the mapping equations was conducted on the region 

 from Matagorda Bay to Apalachee Bay (Figure 22) . This region is the 

 result of joining Figures 3 and 4 and represents in this study, the 



45 



