144 BELL SYSTEM TECHNICAL JOURNAL 



may be considered and are indicated by the network shown in the 

 figure in fine lines. On one side of the curve x is positive and on the 

 other it is negative. Consider the equation 



w{z) -1 = 



and what happens to it as A increases from a very small to a very large 

 value. At first the locus x = lies wholly to the left of the point. 

 For this case the roots must have x < 0. As A increases there may 

 come a time when the curve or successive convolutions of it will sweep 

 over the point w = \. For every such crossing at least one of the 

 roots changes the sign of its x. We conclude that if the point w = 1 

 lies inside the curve the system is unstable. It is now possible to 

 enunciate the rule as given in the main part of the paper but there 

 deduced with what appears to be a more general method. 



Appendix II 



Discussion of Restrictions 



The purpose of this appendix is to discuss more fully the restrictions 

 which are placed on the functions defining the network. A full 

 discussion in the main text would have interrupted the main argument 

 too much. 



Define an additional function 



<^^ = T-- r^T^'^^^'^^' - ^ <x<0. (63) 



Zirt J IK — Z 



n{iy) = lim n{z). 



This definition is similar to that for iv{z) given previously. It is shown 

 in the theorem ^^ referred to that these functions are analytic for 

 X 5^ if AJ{iu)) is continuous. We have not proved, as yet, that the 

 restrictions placed on G{i) necessarily imply that J{iw) is continuous. 

 For the time being we shall assume that J{iui) may have finite dis- 

 continuities. The theorem need not be restricted to the case where 

 J{io}) is continuous. From an examination of the second proof it will 



be seen to be sufficient that I J{iui)d(iw) exist. Moreover, that proof 



X 



can be slightly modified to include all cases where conditions (AI)- 

 (AIII) are satisfied. 



" Osgood, loc. cit. 



