266 H. K. HARTLINE, F. RATLIFF AND W. H. MILLER 



each receptor in the other group. In each group the r's of the receptors are 

 equal and two simultaneous equations suffice: 



/•a = ^A - [(«A - ^aaCz-a - '-Va) + "b ^abO-b - '■\^)] (6) 



^B = ^B - ["a ^Ba(''a - '■"ba) + («B - ^Bb(''b " ''"bb)] 



The validity of these equations is established by solving the entire set of 

 "a + "b equations (4), subject to the idealized assumptions. The notation 

 used in these equations has the following meaning: the two groups are desig- 

 nated A and B, containing /7^ and iIq receptor units, respectively. They are 

 illuminated at intensities such that any single receptor in group A if illumin- 

 ated by itself would have a frequency e^; in group B, e^. When the two groups 

 are illuminated together, each receptor in group A responds at a frequency 

 r^; in group B, at r^. A'^a is the coefficient of the inhibitory action between 

 any two receptors in group A, and r%A is the threshold of that action. Like- 

 wise Kq^ and /-OgB are the inhibitory parameters for the interaction within 

 group B. A'ab is the coefficient of the inhibiting action (/'Vb its threshold) 

 exerted by each unit in B on each unit in A; Kq^ and r%A ^re the parameters 

 of the action exerted by receptors in A on receptors in B. Collecting terms and 

 dividing the first equation by \ ^ (n^ — \) K^^ and the second by 

 1 + ("b ~ ^bb w^ have 



^A = ^A - ^ab(''b - '■\b) (6a) 



^B = ^b - ^ba(''a - /""ba) 



where we define 



(". 



1+ («A - 1) /^AA 



e^ likewise (6D) 



f. _ ^'B ^^ab ^ Hkewise 



^° l + K-D/^AA ^^ 



We note (cf. equation (5)) that e^ is the frequency we would obtain from 

 each unit in A by illumination of the entire group A alone, at an intensity 

 that would yield f^ if restricted to any one of the units in the group; likewise 

 e^. The parameters A^ab and K^^ may be considered coefficients of group 

 interactions (respectively, group B on group A, and group A on group B). 

 Thus the two groups may be treated as units, each with a certain amount of 

 self-inhibition, acting on each other with coefficients that depend on the 

 numbers of ommatidia involved. 



The variation of group inhibitory coefficients with group size has been 

 examined experimentally. Figure 18 shows that enlarging a group increases 



