A MATHEMATICAL THEORY OF RANDOM MIGRATION 5 



of Sciences of Amsterdam, entitled "A local probability problem. "* Professor 

 Kluyver obtains the general solution in terms of the integral of a product of 

 Bessel's functions of the zero and first orders. He deduces Rayleigh's solution for 

 n large, he shows that the Bessel function integral represents a series of different 

 analytic functions in different intervals, and proves a number of special problems 

 of very considerable interest. Referring to his general solution, he writes, however : 



" From this result we infer that the probability sought for is of a rather intricate 

 character. The n + 1 functions J are oscillating functions, and have their signs 

 altering in an irregular manner as the vai'iable u increases. Hence even an 

 approximation of the integral is not easily found, and as a solution of Pearson's 

 problem it is little apt to meet the requirements of the proposer."! 



Kluyver's solution is of extreme suggestiveness for the analytical theory of 

 discontinuous functions. In the endeavour to express it in a form suited to my 

 special purposes I have come across a long series of Bessel function properties, 

 some at least of which seem to me novel, but have unfortunately no bearing 

 on the problem of migration. If we turn to Rayleigh's solution for n large, I 

 must confess at once to being unconvinced of the adequacy of the proofs used to 

 deduce it, especially that in the Theory of Sound\. Kluyver's proof of Rayleigh's 

 solution § appears to me to also require much strengthening, and in neither case do 

 we have any practical measure of what the number of flights must be before we 

 have in practice a reasonable accordance between the discontinuous Bessel's function 

 integral expression and the Rayleigh solution of Gaussian frequency type. 



After a good many failures I have succeeded in obtaining a solution in series 

 of the Bessel function integral, but this not of a character to be of service for 

 frequent arithmetical calculations. It serves, however, to test the approximation 

 of the Rayleigh solution and the accuracy of the solutions for few flights obtained 

 by other processes. At this stage I realised that the functional equation between 

 the distributions for n and n + 1 flights could be solved graphically, and that starting 

 with the known distributions for n = 2 or n = 3, we could by very great labour, 

 but absolutely straightforward graphical work and the use of mechanical integrators, 

 build up in succession the solutions for n — 4, 5, 6, 7. etc. I proposed that this 

 process should be continued until the graphically found distribution coincided with 

 the plotted values obtained from the above solution in series. This was achieved 

 for n = 7. For n = 6 and n = 7 , the solution in series approaches to the Rayleigh 

 solution, with which for all practical purposes it may be asserted to coincide for 

 71 = 10. We have thus reached a continuous graphical illustration of the transition 

 of a series of discontinuous and, in many respects, remarkable analytical functions, 

 step by step with the increase of n into a normal curve of errors. The relation- 



* Koninklijke Akademie van Vetenschappen te Amsterdam. Proceedings, Oct. 25, 1905, pp. 341 — 50. 

 t loc. cit. p. 343. { 2nd Edition, § 42 a. § Kluyver, loc. cit. p. 345. 



