Antigenic specificity 213 



III. EXPERIMENTAL TESTS 



1. Occurrence of the Heterophile Reaction 



The incidence of heterophile reactions will depend on the number and 

 relative frequencies of the various antigenic determinants. As nothing is known 

 so far about relative frequencies, we assume them to be equal; this will yield 

 a lower bound of the number of different G's. Under the assumption of equi- 

 probability, we use the following argument (4): Let Q and C_, be different and 

 (as far as known) unrelated antigen complexes; let m^ and nVj denote the number 

 of antigens in the complexes which can be differentiated and demonstrated, 

 by a given technique, by reaction with the specific antisera S^ and Sj\ let 

 hij be the number of heterophile reactions observed ; let A^ be the total number 

 of different antigenic determinants which this technique will differentiate. Then, 

 the maximum likelihood estimate N of N is given by: 



"a 



Assuming ft to be one, we have: 



H(G) ^ logo N 

 and 



This is a preliminary test of r'^'^. 



2. Classification of Cross Reactions 



The strength of the cross reaction presumably depends on the number of 

 letters in common, and on the nature of these letters. We assume as a working 

 hypothesis that the former factor is the leading one. Then, if we grade the 

 strengths of many cross reactions, we expect to find a distribution into clearly 

 separated groups such as strong, less strong, weak, . . . etc., cross reactions. 



We suspect that the strong cross reactions are those in which the G-pair 

 has (^ — 1) letters in common, the next class those with (k — 2) letters, and the 

 weakest observable reactions those with one common letter. Then, the number 

 of distinguishable classes of strengths of cross reactions should be (k — 1). 



This may develop into a test of k. 



3. Ratio of Incidence of Heterophiles to Incidence of Strong Cross Reactions 



By our hypotheses, the probability of occurrence of heterophiles is the 

 probability of having all letters in common; now, for k letters, assuming p to 

 be one, the number A^ of different words is: 



TV = r'^'^. 



Then, probability of a given word = {\jrY^' 



and, probability of a heterophile = (1//-) 



15 



<xk 



