34 



of high order should be used, say n = l6. For this reason the procedure that 

 has been illustrated in detail may be less tedious for manual application. 



COMPARISON WITH KARMAN AND KAPLAN METHODS 



In order to compare the accuracy of the Ka'rma'n method with the pres- 

 ent one, the error function ^ k (x) was computed for a distribution derived by 

 the Ka'rman method, employing 14 intervals extending from -O.98 1 x ^ O.98. 

 ^ k (x) is graphed in Figure 3- It is seen that the errors are much greater 

 than for the present method, especially near the ends of the body. The oscil- 

 latory character of ip k (x) is imposed by the condition that the stream function 

 should vanish at the center of each interval. It is conceivable that the 

 amplitude of the oscillations in ^ k (x) may remain large even when the number 

 of intervals is greatly increased; i.e., the Karman method may give a poorer 

 approximation when the number of source-sink segments is greatly increased. 

 The pressure distribution obtained by the Ka'rma'n method is graphed in Figure 5. 



Kaplan's first method 13 was also applied to obtain the pressure dis- 

 tribution. Kaplan expresses the potential function <t> in the form 



^= XA nV X) P n^ 



where X and n are confocal elliptic coordinates, 



P n U) and QjJX) are the nth Legendre and associated Legendre 

 polynomials, and the 



A n 's are coefficients to be determined from a set of 

 linear equations which express the condition 

 that the given profile is a stream function. 



In the present case it was assumed that was expressed in terms of the first 

 9 Legendre functions and the A n 's determined from the conditions that the 

 stream function should vanish at 9 prescribed points (including the stagnation 

 points) on the body. The resulting pressure distribution is also shown in 

 Figure 5- 



SOLUTION BY APPLICATION OF GREEN'S THEOREM 

 GENERAL APPLICATION TO PROBLEMS IN POTENTIAL THEORY 



Let <f> and u> be any two functions harmonic in the region exterior to 

 a given body and vanishing at infinity. Then, a consequence of Green's second 

 identity 27 is 



JF^-JHS" (85, 



