94 PREVENTIVE MEDICINE 



the truth. If we suppose that the whole average life of the insect 

 contains n stages, and that each insect can traverse an average 

 distance I during one such stage or element of time, then the ex- 

 treme average distance to which any insect can wander during 

 the whole of its life must be nl. I call this the limit of migration 

 and denote it by L, as it becomes an important constant in the 

 investigation. It will then be found that the numbers of insects 

 which have succeeded in reaching the distances nl, (n-- 1)1, (n 2)1, 

 etc., from the centre will vary as twice the number of permuta- 

 tions of 2n things taken successively, none, one, two, three at a 

 time, and so on - - that is to say, as the successive coefficients of 

 the expansion of 2 zn by the binomial theorem. Suppose, for con- 

 venience, that the whole number of gnats escaping from the box 

 is 2 2n - - a number which can be made as large as we please by 

 taking n large enough and / small enough then the probabil- 

 ities are that the number of them which succeed in reaching the 

 limit of migration is only 2; the number of those which succeed 

 in reaching a distance one short stage of this, namely, (n 1)1, is 

 2.2w; of those which reach a stage one shorter still is 



2n(2n 1) 



-JT 



and so on. Hence the whole number of gnats will be found arranged 

 as follows : 



Distance from centre nl (n 1)Z (n 2)1 (n 3)1 etc. total. 



Number of gnats 2+4n + 2 2n(2n-l) +g 2n(2-l) (2n-2) + etc =2 



It therefore, follows from the known values of the binomial 

 coefficients that if we divide the whole number of gnats into groups 

 according to the distance at which their bodies are found from the 

 box, the probabilities are that the largest group will be found at 

 the first stage, that is, close to the box, and that the successive 

 groups, as we proceed further and further from the box, will be- 

 come smaller and smaller, until only a very few occur at the ex- 

 treme distance, the possible limit of migration. And the same rea- 

 soning will apply to a breeding-pool or vessel of water. That is, the 

 insects coming from such a source will tend to remain in its imme- 

 diate vicinity, provided that the whole surrounding area is uni- 

 formly attractive to them. 



The following diagram will, I hope, make the reasoning quite 

 clear. 



We suppose that 1024 mosquitoes have escaped during a given 

 period from the central breeding-pool P, and we divide their sub- 

 sequent life into 5 stages --the numbers 1024 and 5 being selected 

 merely for illustration. Rings are drawn around the central pool 

 in order to mark the distance to which the insects may possibly 



