These equations will be recognized as similar to Fredholm integral equations 

 of the first kind since ^ = ^ {(^ , '^)- As if this were not bad enough, the kernel 

 function, while symmetric with respect to direction (i .e. , K (o-,\//) = K (a, -^), 

 is not symmetric in the more general sense of K (o" , i//) = K {^,°' )o The theory of 

 this type of equation is not well developed. It may be surmised though, that even 

 if the non-symmetric kernel does possess real, non-zero eigenvalues, the best that 

 can realistically be hoped for in a solution to the equation is a type of mean value 

 for H (<'") in a specified °" - interval o This seems perfectly alright, however, since 

 E ( u») is only an estimate in a similar frequency interval inoi - space. The solution 

 then will represent a smoothed approximation to the exact answer. 



The problem of solving (7a) or (8a) can be considered as a generalization of 

 the problem of solving a set of n-algebraic equations in n-unknowns. With this idea 

 in mind (7a) may be approximated as follows: 



E («^ .) = / H (0-) [K (0-,'^)+ K (q-, -^)] do- 

 ^^ °" ^V Isiny/, I 



g 



=E H (o-y Tk (o-j, <// j.)+K (o-^., -v^.yl Scrj 



(9) 



J - 1 



J 



sin \^ .. 

 'J 



n 



g 



A.. H (o-.)„ 

 U J 





Note that the finite difference representation is general enough to allow for unequal 

 intervals of c j. in this notation^o" ; and w ] represent individual members of a dis- 

 crete set of mid-interval frequencies, while ^ ;■ is the value of ^ determined by 



\|/ i j = cos 



-1 



(or . + (JJ \ 

 J ■ 



^2 V 



J 



The idea is to fix upon values of to j and a :, and with v// y from above, evaluate 

 the coefficients Ajj. For the values of cr • and oj j for which ^ -S(^ <0<^ + 

 S^// / Aj: must be evaluated from (8a) due to the singular nature of (7a) at »// = 0. 

 In this case H (a ) varies but slightly over the interval V' equals to8</' and so may 

 be considered as constant in that range (Cartwright, 1963). Hence, (8a) is approximated 



28 



