198 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



l/lOn — 



APi,{l + kf 



B 



APl2 \ 



B 



k' 



fc(i - r) 



kj 1 - F 



Mnil - k")(k - k") 



(1 - ky 



If the initial series is a complete series, n -^ =o and 



(q) 



?/lOo 



t/leoo 



APn(i-\-ky 



B{1 - ky L B 



'APi2 k(l - k) 



1 + k 



- Mnk 



1 + k 



'2APi2 kil - k) 



B 



(1 +/c) 



- Muil + k) 



(22) 



C. Partial Front Series 



The worst rebound occurs when heel and front impacts occur nearly 

 simultaneously, with the front hitting first. From Equation (m) for an 

 initial front series, this requires that 



B 



APu 



= Q = 



1 - A;" 



k - k- 



(30) 



After n front impacts conditions are given by Equations (14) and 

 (19) with yi = tjt = 0, and 





= 1 - (1 - Mx2)(l - kn = k'" + 



,„ , Mnyl 2Mi2A:"r/2 



Ph 



This may be solved for y-z = Pi2(l — k"). The maximum front excur- 

 sion now possible is that for a complete series of heel impacts. The 

 above value of ?/2 in Equation (24) yields 



2CYi = Mu + (1 - Mi2)[/c'" - ilfi2(l - fc")'] 



(31) 



D. Partial Heel Series 



The worst rebound occurs again when heel and front impacts occur 

 nearly simultaneously, with the front hitting first. From Equation (9) 

 for an initial heel series, this requires that 



B 



APn 



= Q = 



1 



k{l - k) 

 1 + k 



+ k{\ - k^)Mi 



(32) 



