BOUNDARY PROBLEMS AND DEVELOPMENTS. S9 



(51) z>(\) = i [« ( »>] c + [««] 5:; e * r ^ +*cw |. 



Factoring from the determinant the exponential factors, any one of 

 which occurs in each element of an entire column, the further alterna- 

 tive form 



(52) D(\) = n e ,xr * (6) +Bk(b) ]8tk I [«#] 8* cc + [«&>] 5 



&-i 



** 



cc 



is obtained. It becomes necessary at this point to differentiate be- 

 tween certain types of conditions which inherently characterize any 

 particular system of type (46). 



The conditions of the system will be said to be regular (i) if n = 1, 

 or if, when n = 2, arg Vjib) 4: arg { =>= T{(b) } when j =1= i, and (ii) the 

 boundary conditions are such that each of the determinants 



(53) W™ = | «,« 5<? + „« *2» 



differs from zero. 



The conditions of the system will be called irregular if either of these 

 conditions is not satisfied. In the further discussion we shall consider 

 first the case of regular conditions and then the particular type of 

 irregular case which results from dropping the regularity condition (i) 

 above. 



Case I. Regular Conditions. In this case the rays i2{xr"i(6)} = 0, 

 i = 1, 2, . . .n, are distinct and divide the plane into 2n sectors of 

 type °"w Let us fix the attention upon any one of these rays, say 

 the ray R{\T v (b) } = 0, and let %„ and c„ T denote the abutting sectors, 

 the former being that within which R{\T v (b)} < 0. For X on the 

 ray i?{Xr„(6)} = the v column of D(\) consists of terms whose 

 form cannot be abbreviated by means of relations (50). Retaining 

 therefore the original expressions for the elements of this column we 

 have 



* ** 



£(x) = I [«#] 8^ + ka {ft? + 8 C „\ +T.W+W i . 



For X either within a^ or within o VT , on the other hand, the form of 

 Z)(X) is given by formula (51), the difference in the value of D(X) in 

 these sectors being accounted for by the difference in the values of 

 8 CC and 8* c l, c = 1, 2,. . ., n. 



