BOUNDARY PROBLEMS AND DEVELOPMENTS. 95 



hypothesis) it is seen that the characteristic equation, i.e. D(\) = 0, 

 is in each case of the form 



(69) 1 + N, xr "' (6) + [c 2 ] / xr *' (6) + N *< 1+ 'l* r '.<*> = 0, 



where Ci, i = 1,2, 3, are complex constants, c 3 ^ by hypothesis, and 

 r is a real constant r = 1 . 



Wilder n has shown that the roots of this equation are asymptotically 

 represented by those of the equation 



(70) 1 + d 6> xr <" (6) + c 2 / xr » (6) + c. e\ 1+r ! xr "' (6) = 0, 



and has discussed the distribution of the roots of this equation. We 

 shall proceed to this discussion, observing, however, first that when 

 | r„ (b) | and | r„ (b) \ are commensurable a far simpler treatment is 



V 



possible. In that case r = — where y and q are integers and the 



equation (70) is an algebraic equation of degree (p + q) in e q 

 Accordingly it has (p + q) roots, i.e. 

 Ar„,(&) 

 e a =«,-, J= 1,2,..., (p-{-q) 



from which it follows that 



(71) X* = j^{log« ? - + 2M. 



In this case, therefore, the characteristic values which lie in sector S 

 for | X | > N are asymptotically represented by a set of points which are 



2qv 

 spaced at intervals of length , r /, , , on (p + q) lines (not neces- 

 sarily distinct) parallel to the line .ftfXr^ (b)} = 0. 



When r„ 2 (fr) and T n (b) are incommensurable no such simple treat- 

 ment is possible. The distribution of the characteristic values may 

 be obtained, however, by Wilder's procedure, 12 which follows. 



Setting Xi\i(6) = z = x + iy we have as the equation (70) 



/(*)= 1 + c lC * + c 2 e rz + c 3C l 1+r l 2 = 0, 



11 Wilder, loc. cit., p. 423. 



12 Cf. Wilder, loc. cit., pp. 420-422. 



