120 



BIRKHOFF AND LANGER. 



and substituting for D(X) its value as given by equation (56) we have 



n 

 7T 



k-1 



-\ \TAb)+BAb) | fiM Dji(\) 



z>(x) 



where Z)(X) is given for the case of regular conditions by formula (57) 

 and for the case of irregular conditions discussed in section VII by 

 either formula (66) or formula (68). A glance at these formulas 

 shows, however, that in each case the expression for D(\) reduces, 

 under the conditions for which the expression is valid to the form 



£(X) = W^ + <p 2> on arc C' M „. 



Now every point of a circle C, and hence in particular of an arc C^ 

 is at a distance which exceeds a fixed quantity from any characteristic 

 value. However, by analogy with formula (59), for any point of C^ 

 under regular conditions for which the function D is sufficiently 

 small we have 



In other words the point in question will necessarily lie near one 

 of the characteristic values. 



This stands in contradiction with the fundamental property of the 

 circle C. 



Inasmuch as a similar relation holds on every arc of type C^, 

 it follows that for every point of all circles C, D(\) exceeds, under 

 regular conditions, some fixed positive constant 5i, i.e. 



(114) 



D(k) |= <h > for X on any circle C. 



While we have thus proved the relation (114) only for the case of regu- 

 lar conditions it can be shown to hold equally well under the type of 



