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BELL SYSTEM TECHNICAL JOURNAL 



We substitute the special values x = pt,y = qt after obtaining the expansion. 

 Let 



^1 = Q/P, h = -b/P (1.6) 



The most general conditions of interest are comprised in the ranges: 



0<y^i<l, -2<^o<2'* (1.7) 



To P 



Fig. 4. — Regions in x3'-plane bounded by ^o + cos x ■\- k\ cos )» = 0. 



The regions in the x^-plane in which g{x,y) does not vanish are bounded 

 by the various branches of the curve : 



^0 + cos :v + ^1 COS T = (1 .8) 



We need to consider only one period rectangle bounded by x = ±x, y = zLir, 

 since the function repeats itself at intervals of lir in both x and y. The 

 shape of the curve (1.8) within this rectangle may have three forms, which 

 are depicted in Fig. 4. In Case I, ^o -\- ki > k, ko — ki < 1, the curve 

 divides into four branches which are open at both ends of the x- and y-axes. 

 In Case (2), )^o + ^i <1, ^o — ^i > —1, the curve has two branches open 



