Decembeb 1, 1905.] 



SCIENCE. 



693 



have wandered more or less backward and 

 forward and have died in the vicinity of 



Distance from center 7il {n — 1)1 (n — 



2ii{2n 



Number of gnats 



2 + in +2 



and so on. Hence the whole number of 

 gnats will be found arranged as follows: 

 2); {n — 3)l etc. total. 



1)(2m — 2) 



— 1) ^ g 2n(2ra 



3! 



+ etc. 



the box or pool from which they originally 

 came. 



The full mathematical analysis determin- 

 ing the question is of some complexity ; and 

 I can not here deal with it in its entirety. 

 But if we consider the lateral movements as 

 tending to neutralize themselves, the prob- 

 lem becomes a simple one, well known in 

 the calculus of probabilities and affording 

 a rough approximation to 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 extreme 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 center will vary as twice the num- 

 ber of permutations of 2n things taken suc- 

 cessively, none, one, two, three at a time, 

 and so on — that is to say, as the successive 

 coefficients of the expansion of 2^^ by the 

 binomial theorem. Suppose, for conve- 

 nience, that the whole number of gnats es- 

 caping from the box is 2~'^—Si number 

 which can be made as large as we please by 

 taking n large enough and I small enough 

 — then the probabilities are that the num- 

 ber of them which succeed in reaching 

 the limit of migration is only 2 ; the num- 

 ber of those which succeed in reaching a 

 distance one stage short of this, namely, 

 {n—l)l, is 2.27^ of those which reach a 

 stage one shorter still is 



2»(2rt — 1) 

 ^ 2l 



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 become smaller and smaller, until 

 only a very few occur at the extreme dis- 

 tance, the possible limit of migration. And 

 the same reasoning will apply to a breeding 

 pool or vessel of water. That is, the in- 

 sects coming from such a source will tend 

 to remain in its immediate vicinity, pro- 

 vided that the whole surrounding area is. 

 uniformly attractive to them. 



The following diagram will, I hope, make 

 the reasoning quite clear. 



Diagram I. The chance-distribution of Mos- 

 quitoes. P, central breeding-pool. L, limit of 

 migration. The numbers denote the proportions 

 of 1.024 mosquitoes starting from P which die at 

 the distances 1, 2, 3, 4, 5, respectively. The 

 continuous line denotes a continuous migration 

 always in one direction; the dotted line, the 

 usual erratic course. 



