116 John B. Calhoun 



value of N at the inflection point of /^^, that is at the point where the 

 second derivative of /„<, as a function of N is zero, it may be shown that 



Ni= \+— (81) 



abHo 



Then inserting Eq. (77) into Eq. (81) 



;,,.!+ ^J^hjZ^ (82) 



Mo 

 Then inserting Eq. (80) into Eq. (82) 



Ni= \ + 2(No - 1) (83) 



Where A^ is Ni, Eq. (83) becomes 



iV, = 1 + 2(iV6 - 1) - (84) 



= 2N,- I (85) 



B. Basic Processes Involved in Social Interaction 



1. Satisfaction and Frustration as a Function of Group Size 



Satisfaction and frustration from social interaction are by definition 

 measured, respectively, by da and 6/. Full satiation attains at d^""^ and 

 optimum frustration at dj"'\ These equivalent cjuantities are equally 

 necessary for the individual to persist in an optimum state. We shall here 

 be concerned with the effects upon 6a and df resulting from varying A'' 

 when ab, fib, and fxb remain constant and appropriate to Nb. It must be 

 recalled from the statement preceding Eq. (65) that df''^ is not used in 

 the sense of the maximum af'^p but rather as the amount of time spent in 

 frustrating refractory periods at iV„t, that N where the maximum amount 

 of time, 0^J^\ is spent in satisfying refractory periods. As implied in Table 

 XI, B and Fig. 35, af'^p attains a maximal value when A^ = infinity. 



As Nb increases i and a must decrease in order to maintain d^'"^ [see 

 Eqs. (56), (61), and (69-71)]. Likewise, for any arbitrary series of a, 

 such as 1.0, 0.75, 0.5, 0.25, 0.1, 0.05, 0.025, and 0.01, there must be respec- 

 tive Nb at which a/^^^ = 0f \ Each such a with its corresponding Nb 

 might be considered as representing a distinct species. For each species 

 circumstances may cause N to diverge from its Nb. In any such divergence 

 da diminishes, and for the species where a remains constant, /„« will exhibit 

 changes proportional to da. Thus, faa may be taken as an index of the degree 

 to which changes in A^ from Nb diminish satisfaction from social interac- 



