570 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



The derivative of Q{p) vanishes at 



IHn = HI + ^/*); q[l^ =- HI - d*) (A26) 



Avhich gives the symmetric configuration. Clearly this value of p gives 

 to Qiv) its maximum value, ^1 — d*^). This proves that the symmetri- 

 cal configuration is least favorable in the limit as n -^ oc . 



Under the configuration (Al) with d = (/* and n -^ co the distribution 

 of the chance variables Zi {% = \, 2, ■ ■ ■ , k — \) approaches a joint 

 multivariate normal distribution with zero means, unit variances and 

 correlations given by 



piZ.Z,) = , , ^''''^''' — y^ a ^ j) (A27) 



?>[i]9[i] + {Vm - d*){qii] + d*) 



which do not depend on n. For the symmetric configuration this reduces 

 to the simple form 



p{ZiZj) = i a y^j). (A28) 



This is precisely the case which arises in [1] and consequently the tables 

 in [1] can be used for our problem when (the answer) w is large. The con- 

 stants C = C(P*, k) tabulated in [1] solve the equation 



P Izi >-:^{i = l,2,---,k-l)\ = P* (A29) 



for standard normal chance variables Zi satisfying (A28). If we equate 

 C/s/2 and the corresponding member of (A22), then we obtain for the 

 symmetric configuration 



X.^ d*Vn 



V2 — /i " (A30) 



or solving for n and letting B = jC this yields the large sample normal 

 approximation 



n^-^Jl-d*') (A31) 



Since d* is usually small when n is large and since the solution in (A31) is 

 usually somewhat smaller than the true value, then it is of interest to 

 examine the simpler approximation 



n ^ ^^ (A32) 



which is greater than the result in (A31). This is called the straight line 

 approximation since it plots as a straight line on log-log paper as shown 



