112 



BIRKHOFF AND LANGER. 



«!?(*)• = 1-4 / I Y(x)(6*i)Z(t)R(t)F(t).dtd\ 



c im 



O a 



S%(x)- = L~ I f Y(.v)(dZ)Z(t)R(t)F(t)-dtd\ 



C SlTl 



(101) \ 



Cpv 







Sff(*)-= 1^4 f TWA- ] fw m Y(a)QfS 



n V.irl *J *J 



)Z(t)R(t)F(t)-dtd\ 



C &TI 



Ciiv 



b 



&x)- = L^~. fY(x)A-i fjV b Y(b)(8*j)Z(t)R(t)F(t)-dtd\ 

 c 2-kiJ ° 



C M „ denoting any arc of circle C which lies within the sector a ^ and, if 

 n ^ 2, upon which none of the quantities R {X{y t (a;) — 7 3 -(a:)} change 

 sign. If C^ abuts upon a ray bounding o ^ it shall either include or 

 exclude the end point for which the quantity R {XT^b) } or the quantity 

 R {\r„(7>)} vanishes according as the quantity in question is < or 

 > within c The arc C uv may, therefore, include one, both, or 



neither end point, and since the reasoning is precisely the same in each 

 case (the sector can be split if both end points are included) we shall 

 consider for the sake of concreteness that it includes one, namely that 

 one which lies on the ray R {\T„(b)\ = 0. The symbol 2 indicates 



that the sum of the integrals over all arcs composing the circle C is to 

 be taken. 



We shall proceed to evaluate each of the integrals above in turn, 

 and in the course of this evaluation it will be convenient to refer to the 

 facts established by the following lemmas. The notation | (p{x) \<M 

 will be used to indicate that the function in question is bounded. It 

 is not to be understood that the M in any one case represents the same 

 constant as in any other, but merely that there exists in each case 

 some constant for which the relation is true. 



Lemma 1 : Given any function <^i(.r, X) which is such that 



(i) | Vl (x, X) | < M for a^ x ^ 13, | X | > N , 



(102) 



(ii) lim I <pi(x, X) | = uniformly, for a ^ x ^ j8 — x< 



x = » 



