4 KARL PEARSON 



(ii) The species scattering from a centre is supposed to distribute itself 

 uniformly in all directions. The average distance through which an individual of 

 the species moves from habitat to habitat will be spoken of as a "flight," and there 

 may be n such " flights " from locus of origin to breeding ground, or again from 

 breeding ground to breeding ground, if the species reproduces more than once. 

 A flight is to be distinguished from a "flitter," a mere two and fro motion associated 

 with the quest for food or mate in the neighbourhood of the habitat. 



(iii) Now taking a centre, reduced in the idealised system to a point, what 

 would be the distribution after n random flights of N individuals departing from 

 this centre ? This is the first problem. I will call it the Fundamental Problem of 

 Random Migration. 



(iv) Supposing the first problem solved, we have now to distribute such points 

 over an area bounded by any contour, and mark the distribution on both sides 

 of the contour after any number of breeding seasons. The shape of the contour and 

 the number of seasons dealt with provide a series of problems which may be spoken 

 of as Secondary Problems of Migration. 



A little consideration of the Fundamental Problem showed me that it presented 

 considerable analytical difficulties, and I was by no means clear that the series of 

 hypotheses adopted would be sufficiently close to the natural conditions of any 

 species to repay the labour involved in the investigation. At this stage the matter 

 rested, until last year Major Ross put before me the same problem as being of 

 essential importance for the infiltration of mosquitoes into cleared areas, and asked 

 me if I could not provide the statistical solution of it. He considered that we 

 might treat a district as approximately "equi-swampous," and thus my conditions 

 (i), (ii) above could be applied to obtain at any rate a first approximation to 

 the solution. 



Starting on the problem again I obtained the solution for the distribution after 

 two flights, an integral expressing the distribution after three flights, which I 

 carelessly failed to see could be at once reduced to an elliptic integral, and the 

 general functional relation between the distribution after successive flights. At this 

 point I failed to make further progress, and under the heading of "The Problem 

 of the Random Walk " asked for the aid of fellow-mathematicians in Nature*. The 

 reply to my appeal was threefold. Mr Geoffrey T. Bennett sent me in terms of 

 elliptic integrals the solution for three flights. Lord Rayleigh drew my attention 

 to the fact that the " problem of the random walk " where the number of flights 

 is very great becomes identical with a problem in the combination of sound ampli- 

 tudes in the case of notes of the same period, which he has dealt with in several 

 papersf. Lastly Professor J. C. Kluyver presented a paper to the Royal Academy 



* July 27th, 1905. 



+ Phil. Mag., August, 1880, p. 75; February, 1899, p. 246. 



