24 BELL SYSTEM TECHNICAL JOURNAL 



The average quality after inspection {Pa), after substituting in equation (12) 

 the value of / given in equation (8'), is obtained from the relation 



Differentiating with respect to p in accordance with equation (14) gives, 

 dpA (N - n) r^^ e-P^ipn)"" e-P"{pny+^ 



dp N 





(16) 



Equating to zero and solving for p, gives the value oi p = p\ that makes Pa 

 a maximum; i.e., pA = Pl- 



Let pin = x; the particular case covered by equation (15) where p = pi, 

 and Pa = pL may then be expressed as 



N -n "^ e-'^x"' 



Nn ^0 m\ 



or 



?.-yg-i), (18) 



where 



m=c g— i„m 



Similarly, equation (16) equated to zero becomes, after substituting p^n = x 

 and simplifying, 



E — \ r— = 0- 20 



Substituting in equation (19) the second term of equation (20) for the sum- 

 mation term gives 



y-^r- (21) 



These relations* provide a basis for determining the values of x and y, cor- 

 responding to specific values of c, listed in Table A. The values of x for c = 

 to 30 were determined from equation (20) using Newton's Method of Approxi- 

 mation. The values of x for c = 31 to 40 were estimated on the basis of successive 

 differences. The listed values of y are averages of the two values determined 

 from equations (19) and (21), which differ slightly because values of x were 

 determined to only two decimal places. 



* Reduction of the mathematical relations to this simplified form and the determina- 

 tion of several .t and y values, were contributed by Dr. Walter Bartky of the University 

 of Chicago (when he was associated with the Western Electric Co.) shortly after the de- 

 velopment of the AOQL concept and the preparation of preliminary ylOQL double sampling 

 tables. The methods and work of computing the values in Table A were contributed 

 by Mr. George C. Campbell, formerly of the Bell Telephone Laboratories. 



