214 Bernard N. Jaroslow and Henry Quastler 



Since we propose that in the strong cross reaction we have (^ — 1) letters 

 in common, the probabihty of this event is: 



There are k sets of {k — 1 ) letters in a A:-letter word ; hence, 



probability of a strong cross reaction --^a k{\lrY^^~''-^ 

 and 



probability (iieterophile) (l/r)"^" 1 



probability (strong cross reaction) <xk{\lr)^-^^~^^ a.kr'^ 



This is a test of a kr'^. One can construct similar tests for the ratio of strong 

 to weak cross reactions, if experimental results warrant this. 



Optimally we have, thus, three experimental determinations of three para- 

 meters which have computable theoretical values. 



IV. METHODS 



It is essential to this study that the test antigens be as nearly as possible 

 a random sample of natural antigens. Since related organisms regularly have 

 common antigens, we chose our sources so that no two were in the same phylum. 

 We used Guyer (5) as our guide to classification. 



Entire organisms were placed in a Waring blender for two minutes with 

 phosphate buffer (pH 7.4) as the extracting agent. If the antigen source was 

 of microscopic proportions it was ground in a mortar and pestle containing 

 sterile powdered carborundum. The material was centrifuged at low speed 

 to remove the gross particulate material and 0.2 per cent formahn was added as 

 a preservative. 



Rabbits were immunized with a series of four intravenous injections of 

 antigen over a ten-day period. They received another series of three injections 

 a month later and were bled three days afterwards. Serum was collected, 

 complement was inactivated by incubation for one-half hour at 56°C and 

 1 : 10,000 merthiolate was added as a preservative. 



Our test system was double diffusion agar precipitin test based on the methods 

 of OuDiN (2) and Ouchterlony (6). Antiserum (0.5 ml) was placed at the 

 bottom of a test tube (4 mm i.d.). The antiserum was topped with a layer of 

 1.5 ml of 1 per cent agar. After the agar had gelled, the antigen (0.5 ml) was 

 added. Both antigen and antiserum diffused towards each other through the 

 agar, and where they met in the proper proportions a band of precipitate 

 became visible. For antigens with different rates of diffusion separate bands 

 appeared. Thus, the number of bands of precipitate estabhsh a lower limit 

 for the number of antigens in the extract. 



We should be able to differentiate between heterophile and cross reactions 

 because the formation of a precipitate is dependent upon the relative concen- 

 trations of antiserum and antigen. In the region of great antibody excess no 

 precipitate forms, although all the antigen is combined with antibody. In 

 the region of large antigen excess no precipitation is seen, although all the 

 antibody is combined with antigen. Between these two zones is a region in 



