BOUNDARY PROBLEMS AND DEVELOPMENTS. 97 



by loss than Dir as .r varies between the limits .r = —A' and .r = + .V 

 along a line // = constant. 



Because of the relation (,721). however, we know that for every 3 

 on line x = —X,f(z) lies within a circle of radius \ about the point 

 a = lj and hence that as a moves along this line arg/(z) changes by 

 less than 2 sin -1 \. Similarly relation (72ii) shows that argiA~) . u } 

 changes by less than 2 sin" 1 x ;i s - moves along the line .r = A". From 

 the identity 



arg/(z) = arggi 1+r }* + arg -yj^j; 



it follows, therefore, that the increase in arg/(z) as : moves along the 

 line x = X from y — \\ to // = ¥•: lies between (1 + r) j Y% — }\\ + 

 '_' sin -1 x and (1 + r) J }•_> — 1\} — 2 sin -1 \. Consequently the in- 

 crease in arg/(z) as z describes the perimeter pf K lies between 



(1+r) I Y 2 -Yi\ + 67r+4sin- 1 X and (1+r) ( F 2 - }\! -Gtt-4 sin^x, 



and accordingly the number of zeros located in the interior of K must 

 lie between 



(1 + r) [Ys-Yx] 



+3 + - sin- 1 X 



and 



27T 



(l+r){Y 2 -Y 1 ) Q 2 . 

 ■ — o — — sin -1 x 



27T 7T 



Since \ can be chosen arbitrarily small, however, this means that the 

 number of characteristic values between any two lines y = Ci and y — 



Ci+ / is at least / — 3, and cannot on the other hand exceed 



2- 



1 + ', + ;, 



27T 



Summarizing the results it is seen, therefore, that the characteristic 

 values in sector S lie in a strip bounded by two parallels to the line 

 R{\r„ {!>) ! = 0, and that they are so distributed throughout, this strip 



that for I X I > A no more than three lie between anv two lines which 



2tt 



are at a distance (/ < from each other and are perpendicular 



1+r l 



to the line ft{Xr„i(6)j = 0. 



