54 KARL PEARSON 



for practical work until n = 6 or 7. Many other shapes of populated or of cleared 

 areas would provide problems of some interest, especially when the spread of the 

 colony was limited in one or more directions by environmental barriers, such as sea, 

 river or mountain range. The problem of sterile areas has been by no means 

 exhausted, for in such cases I have only dealt with a result of the first 

 migration, but actually there will be a second and later migrations in which 

 not only new immigrants will appear but a portion of the first immigrants will 

 be emigrants and again able to breed when they reach uncleared country. Our 

 solution thus gives only a minimum limit to the percentages if the immigrants 

 do not die at the end of the first breeding cycle. Much interest attaches 

 also to cases in which the fertility and the death-rate are correlated with the 

 density, i.e. jxA is not to be considered a constant. But in these as in other 

 problems which suggest themselves, a further preliminary knowledge of some of 

 the ecological constants suggested by the present enquiry would be an extremely 

 valuable guide to the direction that research should take. 



On the purely mathematical side the problem of the "random walk" may 

 now be considered as fairly completely solved. The distribution curves have been 

 determined until they pass into an analytical solution expressed by a new type 

 of function. The expansion in these functions shows the limits to the accuracy 

 of Lord Rayleigh's solution of a certain allied problem in the theory of sound. 

 But the tu-functions which have arisen in the enquiry have most interesting 

 properties, and have led me to a whole series of allied functions of one and 

 two variables which I propose to discuss on another occasion. The expansion 

 in w-functions will I venture to think be found ultimately to have considerable 

 importance for mathematical physics, especially in the evaluation of certain 

 definite integrals which arise there. The possibility of practically carrying out such 

 expansions depends on the determination of the successive moments (and products) 

 of the original function, a process with which every statistician is now fairly 

 familiar. But applied to definite mathematical functions it loses the disadvantage 

 with which it is burdened in statistical practice — the high relative probable 

 error of very high moments — and becomes closely allied to the process of deter- 

 mining the integral of the product of any function and a Legendre's coefficient 

 (or solid harmonic). Should the generalised co-functions prove, as I anticipate, 

 of some mathematical interest, it will be another illustration of how the need 

 of the applied mathematician has thrust him, almost unawares, into the path 

 of a novel functional development. 



