16 



KARL PEARSON 



small, compels us to use a very great number of co-functions, and the convergency 

 of (xxxiv) is then small. 



Another method of determining the distribution of the dispersed population has 



then to be applied to the case of n small. 



(6) Graphical Solution of the Fundamental Problem for n small. 

 Let us consider the general functional relation (i) 



1 f2ir 



<£„ +1 (r 2 ) = 2^ j Q & (r * + ? " 2H C ° S 6) 4 °' 

 Suppose the graph of <£„ from to nl known. This may be any discontinuous function. 

 From nl to » , it will be zero. Let ABD be the graph of <f> n and OA the axis. 



OP = r. Round P describe a circle of radius /, take the radius PQ, so that the 

 angle OPQ = 0; then clearly, OQ 2 = r 2 + f - 2rZ cos 9; rotate OQ round down into 

 line OD, as ON ; draw the ordinate of the graph Nq, then we have 



JSr q = <l> n (r 2 + l 2 -2rlco8d) 



1 



and 



4>n + ,(OP 2 ) = 



2tt 



2ir 



Nqdd. 



Hence if we divide the circle up into a number of equal parts, and determine the 

 ordinates Nq, corresponding to each of them, we can plot a curve to the base 2v, 

 of which the mean ordinate will be c/>„ +1 (OP 2 ), or the ordinate at r of the new 

 curve of dispersion for n + 1 flights. This can be done for a series of values of 

 r from to n+1 I and thus <f> n+1 (r 2 ) will be determined as a new graph. The area 



