BOUNDARY PROBLEMS AND DEVELOPMENTS. 91 



Since the roots of D(\) =0 and those of D(\) = are the same we 

 have, therefore, as the characteristic equation 



(58) [W^ v) ] + [W {VT) } e xr »(W+«.(» = 



This yields, since W^ v ' =fc 0, the equation 



TJ-(w) 



Wv(b) + B v (b) 



If 



-(">•) 



TW<J*v) 



where e is here introduced as a generic symbol for functions which 

 approach the limit zero uniformly as | X | increases beyond limit. 



Solving for X, and observing that log (K + e) = log K + e, we see, 

 therefore, that every characteristic value which lies in sector S is of 

 the form 



(59) Xp = rib) ( Bv{h) + log ~lr^ + e (Xp) + 2pwi \ ' 



where p is a positive integer. Moreover, since Y(x) is analytic in X 

 throughout the sector S, \ X | > N, the same is readily seen to be true 

 of e also. 



Consider now a small circle of fixed radius r drawn about the point 



for any given p. Then for a proper choice of origin in the X plane 

 (see page 98) this circle lies entirely within the sector cr M „ or <r„ r , and 



| — — | < r for | X | > N. Also if p is sufficiently large the point 



1 \ , , -IV^ ) 



B,(b) + log— ^j + e +2 pm 



T v (b) ( ° W 



lies within the circle for all X on the circumference. Consequently 



arg r ~ t^)\~ Bv{b) + ]og ~^ +^ + 2^| I 



increases by 2ir as X describes this circumference, and just one root of 

 equation (59) is accordingly seen to lie within the circle. Since it is 



