320 BELL SYSTEM TECHNICA L JOURNA L 



G(p,(T)dA over the entire p,o--plane is equal to unity, corresponding to 

 certainty. 



For the sake of subsequent needs of a formal nature, it will now be as- 

 sumed that G{p,(t) = at all points p,o outside of the pi , P2 , ci , a^ quad- 

 rilateral region in the p,o--plane, Fig. 1.1, bounded by arcs of the four heavy 

 curv'es, for which p has the values pi and p2 and a the values ai and ao , 

 with pi and en regarded, for convenience, as being less than p2 and a^ respec- 

 tively. Further, G(p,a) will be assumed to be continuous inside of this 



p+dp P^ 



Pa 

 Pi 



Fig. 1.1 — Diagram of general curvilinear coordinates. 



quadrilateral region, and to be non-infinite on its boundary. Hence, for 

 probability purposes, it will suffice to deal with the open inequalities 



Pi < P < P2, (1.1) ai < a < (T2, (1.2) 



which pertain to this quadrilateral region excluding its boundary; and thus 

 it will not be necessary to deal with the closed inequalities pi ^ p ^ P2 

 and (Ti ^ 0- ^ ao , which include the boundary."* 



' The matters dealt with generically in this paragraph may he illustrated b>- the fol- 

 lowing two important particular cases, which occur further on, namely: 



POLAR COORDINATES: p=|r| = 7?, <r=0 = angle of r. Then p, = A', = 0, 

 P2 = Ri = 'X' , <Ti = di = 0, ffi = $2 = 2ir, whence (1.1) and (1.2) become < R < oc 

 and Q < 6 < lir, respectively. 



RECTANGULAR COORDIN.^TES : p = Re r = .v, <r = Im t = y. Then p, = .v, = 

 — x ^ P2 = X2 = 00, o"! = yi = — =0, 0-2 = vs = «= , whcucc (1.1) and (1.2) become — oo < 

 X < <» and — =»_< y < <«, respectively. 



