THE OPSONIC EXPERIMENT 499 



Let us, for convenience, divide the leucocytes which have 

 been counted into groups, according to the number of organisms 

 they contain ; and let the number of leucocytes in any particular 

 group m be y m . Thus y denotes the number of empty leuco- 

 cytes counted, and y x denotes the number of leucocytes which 

 contain one micro-organism. Now if we compare two records of 

 counts which give the same average, and if collisions occur at 

 random, the distribution of leucocytes amongst the various 

 groups should be the same in the two records, apart from errors 

 of experiment ; and such is found to be the case. Certain 

 workers have adopted the proportion of empty cells, in place of 

 the mean content, as a basis of comparison for obtaining the 

 opsonic index. That there is a relation between the proportion 

 of empty cells and the mean is true ; but to estimate the activity 

 of a community on the basis of the proportion of individuals 

 who have failed to obtain work, is hardly as fair as to compare 

 average work performed. If, however, such a relation exists — 

 that is, if the average content can be calculated from the pro- 

 portion of empty cells — we shall have a method by which an 

 estimation may be made in a few minutes with very little 

 trouble. 



The mathematical treatment is as follows : When a leuco- 

 cyte which contains, say, 5 organisms collides with a free 

 organism, it becomes a member of the group which contains 

 6; and the rate at which such collisions occur is proportional 

 to the number in the group 5. Similarly, an individual in 

 group 6 passes into group 7 on collision, and the rate at which 

 such collisions occur is proportional to the number in group 6. 

 Thus the population of group 6 is increased at a rate pro- 

 portional to the number in group 5, and depleted at a rate 

 proportional to the number in its own group (6); or, for 

 group m : 



(1) ^ 5 = (y.n- I -y )*(t) 



where </>(t) is a complex factor denoting the probability of an 

 ingestion occurring. 



(1) It depends on the number of micro-organisms which are 

 free at the moment, but this is being gradually diminished as the 

 time goes on. 



(2) It depends on the concentration of certain factors in the 

 serum, and this also decreases as the time passes. 



