NON-BLOCKING SWITCHING SYSTEMS 411 



Table II — Crosspoints for Several Values of N 



N^^'^ for N in that equation. The total number of crosspoints required 

 for the seven-stage array is: 



C(7) = {2N''' -If 3N''' + (2N'" - if 2N'" -f {2N'" - 1)2N (4) 



= 36Ar'^' - 46V + 20V'^' - 3V'^' (4a) 



general multi-stage switching array 



Equations (1), (2a), (3a) and (4a) are herewith tabulated as a series 

 of polynomials together with the next polynomial: 



C(l) = N' (1) 



C(3) = 6N'" - 3V (2a) 



C{d) = im'" - UN + 3V'^' (3a) 



C(7) = 36V'^* - 46V + 20V'^' - 3V^'^' (4a) 



C(9) = reV"^' - 130V + 86V'^' - 26V'^' + 3V'^' (5) 



These polynomials can be determined for any number of switching 

 stages from the following formula where s is an odd integer: 



C(s) = 2E; 



s-fl 2k / 



,2F 



2 



8 + 1 



>)^ 



+ N' 



.+1 (2^.+! 



-■)- 



(6) 



An alternative expression equivalent to equation (6) has been sug- 

 gested by S. O. Rice and J. Riordan. The recurrence relation used in 

 individually deriving the foregoing polynomials can be used to directly 

 derive the following formula : 



C{2t + 1) = 



n (2n 



1) 



[(on - 3)(2n - 1)'"' - 2n*] (6a) 



n - 1 

 where s = 2^ -f 1 

 N = n'+^ 

 Table III gives comparative numbers of crosspoints for various num- 



