6 KAEL PEARSON 



ship is a noteworthy one, and not without suggestion for other branches of 

 investigation. 



The exact method of graphical solution will be described later ; the whole labour of 

 it, involving many weeks' work, was due to my assistant, Mr John Blakeman, M.Sc. 



(2) General Analytical Solution of the Fundamental Problem. Let the origin 

 be taken at the centre of dispersion and r be the distance of any small elementary 

 area a from the centre of dispersion. Let <f> n (r 2 ) • a be the frequency of individuals on 

 a after the nth flight, and <£„ +1 (r 2 )a their frequency on the same element after 

 the (n + l)th flight. Let I be the length of the flight. Then only those individuals 

 who were on a circle of radius I round the centre of a after the nth flight can reach a 

 with the (n+l)th flight, and only those individuals of these who take their flight 

 in one definite direction. Let be the centre of dispersion, C the centre of a, 

 P a point on the circle of radius I round C, L PCO — 6, then the frequency per 

 unit area at P is <f> n (r 2 + l' 1 — 2rl cos 0), and the amount which goes in directions 

 between and + 80 is d0/2ir. Hence the frequency per unit area at C after the 

 {n+l)th flight is given by: 



<j> n+l (r>) = ~f\(r° + P-2rlco S 0)d0 (i). 



This is the equation, which I shall speak of as the general functional relation between 

 the densities at successive flights. Now assume : <£„ (r 2 ) = C n J (ur), where C n is any 

 undetermined function of n, I and u, and u is at present an undetermined variable. 

 Then by Neumann's Theorem * : 



00 



Jo (u Jr* + l 2 - 2rl cos 6) = J a (ur) J, (ul) + 2tJ t {ur) J t (ul) cos td. 



i 



1 pr 



Hence : — J C n J (u Jr> + 1" - 2rl cos 0) d0 = C n J, (ur) J, (ul) 



= C n+1 J a (ur), by (i). 



Therefore C n+1 = J (ul) C n . 



It follows that C n = D{J (ul)}'\ where D may be any function of I, but not of n. 



Thus we have : <f> n (r 2 ) = DJ (ur) {J (uT)f, 



where we may sum for any or all values of u. 



Now when n=l, fa (r 2 ) must be zero, for all values of r except r = l to 1 + t, and it 

 then equals N/(2ttIt), t being very small and N the total number issuing from 

 the centre of dispersion. We know, however, that t : 



1 du upf(p) J n (up) J n (ur) dp =f(r), if q < r < p ; 



= 0, it r >p or < q. 



* C. Neumann, Theorie der Besselschen Functionen, S. 65. 

 t Gray and Mathews, Treatise on Bessel's Functions, p. 80. 



