1016 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



we obtain from (2-9) , which we now know to be valid for all L, [ 



(X + s)A{s, t) - I = XB{s, t), (2-11) 



and from (2-10), by recalling that the left side vanishes for L ^ d, 



sB(s, - 1 + X(l - t)B{s, t) = X(l - t)e~^'^^^'B{s, t). (2-12) 

 Hence 



B(«. ') = , + x(i - m - .-<.+».| • (2-13) 



and 



A{s, t) = -4- (1 + XE(s, 0). 



X -f- s 



If we denote the Laplace transforms of Pn{L) and Fn{L) by Pn{s) and 

 /jv(s) respectively, then 



P»W = ;, ^ , _ ,,-,.^»y.. . (2-14) 



and 



jNis) =-^pAs) for .V = 1,2, 

 X + s 



(2-15). 



;g.5 JS'aracf Formula for Fo{L) 



It is possible to solve (2-1) through (2-6) in piecewise analytic form by 

 computing recursively from each interval of length 5 to the next one. We 

 shall obtain the piecewise analytic form for Fo{L) by a direct derivation 

 essentially due to E. C. Molina. 



Suppose k is the number of pattern points which fall into (0, L). Let 

 Xi denote the distance between the i — 1^* point and the i' point (xi is 

 the distance from to the first point) as shown in Fig. 4. The configura- 



Xk 



-®^ ±-zf — a ' ..L ' -^-± 



1 2 L-1 



Fig. 4 — Definition of Xi 



