MATHEMATICAL THEORY OF COMMUNICATION 423 



So that if \i satisfy 



P. = Bai 



Both of the sets of equations for ^t and 7^ can be satisfied since C is such that 



I C-^'' - 8ij I = 



In this case the rate is 



2P,^ilog|^C-^- 

 ~ 2Pi Pi,- 4- 



^Pipij log ^ 



= C - -' 



^Pi Pij tij 



but 



^PiPiMog B, - log Bd = E Pj log ^i - 2P, log Bi = {) 



i 



Hence the rate is C and as this could never be exceeded this is the maximum, 

 justifying the assumed solution. 



{To he continued) * 



