34 KARL PEARSON 



in such a case /jl is not constant for all parts of the integrated area, and we 

 cannot suppose the whole final population to scatter from the original centres. 



If we neglect the i/i 2 , i/< 4 ... terms in (liv) we have the value which would 

 follow from the Rayleigh solution of the fundamental problem, and this can be 

 very readily expressed in geometrical terms. For we mark at once that u x and u m 

 are in type identical curves. Take u x and stretch it vertically in the uniform ratio 

 of /uA m_1 to 1, and horizontally in the ratio of Jm to 1, and it becomes u m . 

 In other words the broken line on Diagram VII. represents the approximate solution 

 in this case after m migrations provided we read 2V(/xA) m-1 for N on the vertical 

 scale and 1 = Jmcr for cr on the horizontal scale. The Table on p. 33 gives the 

 chief results. 



The unit of this table is the length I of " flight." It will be desirable to illustrate 

 its application. Any such application can be of course only a suggestion, and on 

 this account the above Table has been calculated to only a few places of decimals. 

 But such suggestions may not be without value. They will become more than 

 suggestions when our knowledge is greater of the migratory habits of different 

 species. At present only rough approximations can be made as to the values 

 of n and I, and these admittedly are of small weight. 



Illustration I. In captivity I have noted that H. aspersa will live for five 

 years. For two years it does not usually lay eggs, and then it will generally, 

 but not invariably, reproduce twice in the year. This is of course subject to 

 claustral conditions, and while these seem in some cases unfavourable, in others 

 they may be advantageous both in matter of longevity and — owing to the constant 

 food supply — in number of broods. This snail, as far as my observation goes, 

 appears to return to the same shelter after seeking its food. Leaving such 

 "flitters" on one side, I think we might look upon thirty to forty yards as a 

 maximum "flight" for such a snail and regard seven or eight such flights between 

 its egg layings as on the average an exaggeration. 



We might therefore take 1 = 40 yards, n = 8, and an average during life of one 

 brood a year as being quite possible approximations in the case of some snails. 



This indicates that the progress across a boundary into unoccupied country 

 would be such that 1 per cent, of the density at the boundary and, therefore, 

 possibly ^ per cent, of the density in the fully-occupied country, would only be 

 reached at 2061 yards from the boundary after 100 migrations. In other words, 

 such a species would only progress a mile or two at most in a century. Such 

 progress would hardly be noted in any studies hitherto made of distribution ; the 

 limits of a species a hundred years ago were certainly not closely defined to a mile 

 or two, even if they have been recently. Of course there are many other ways in 

 which a slow moving species can be transported than by its own "flights," and 

 further no special stress is laid on the above case, but a study of Table VI. shows 



