92 BIRKHOFF AND LANGER. 



readily verified that every such root is a characteristic value it is seen 

 that (59) determines such a value for every p which is sufficiently 

 large, i.e. that for large values of X the characteristic values lie ap- 

 proximately along a line parallel to the ray E{Xr„(&)} = 0, the dis- 

 tance between two adjacent ones approaching as a limit the finite 

 length 2x/ | T„(b) \, as j X | increases indefinitely. From the deriva- 

 tion of this result it is seen moreover, that a similar sequence of 

 characteristic values lies near each ray i^Xr^)} = 0, and that no 

 further distribution of characteristic values exists. 



Case II. A Type of Irregular Conditions. 10 Let us suppose now 

 that the regularity condition (ii) is fulfilled, that n ^ 2, but that 



(60) argl\(&) =arg{±r j (6)} 



for the pair of values i = v u j = v 2 . ^Ye have then a case of irregular 

 conditions, and while we shall restrict the discussion to the case when 

 the relation (60) holds for only a single set of values i, j, the reasoning 

 to be employed is typical and may be applied with equal success to the 

 cases in which a greater number of the points Ti(b) are collinear with 

 the origin of the complex plane. It is only for the sake of brevity that 

 the simplest, rather than the most general case which results from 

 dropping the regularity condition (i) is treated. 



A review of the discussion applied to the case of regular conditions 

 readily shows that the methods employed there apply equally well to 

 the case in hand and yield the same results in any sector of the X plane 

 which does not contain the line RiXT^ (b)\ = 0. It is, therefore, 

 necessary to consider here only the distribution of characteristic 

 values in a sector S containing this line. Along the line in question 

 neither the expressions for the elements of the v[ h nor of the vf column 

 of D(\) can be contracted by means of the relations (50). Accord- 

 ingly it is found that D(\) takes the form 



* ** 



(61) Z)(X) = | [„«] «« + [«£>] {*« + 8 CPi + 5 C „J c xr c( 6) +fic(6 ) | 



* ** 



throughout the entire sector S, 5^ and 5^ representing the quantities 

 8 CC and 8 CC for X on the ray R{\T n Q>) } = which lies in S. Factoring 



10 For the discussion of a differential system representing a different type of 

 irregular conditions see Hopkins, loc. cit. 



