32 KARL PEARSON 



Now exactly as on p. 26 we may show that : 



^L* dX= k) ^^t & ^- 1)(C/2) 



where : i//^, ( Vm ) = x ,^ ( Vm ) + ^^ . X2(e _ 2) ( Vm ) + ^"^y l ' 3 ' X2(! - 3) ^ 



(s-l)(s-2)(s-S) , v 



Vm = c/('Jmcr), and ^ M is defined on p. 10, Equation (xviii). 

 Thus 



u m = NQ^r' 1 {* (Vm) - \^ m e-^ 2 g, v^ ( Vm ) + ^ 3 v^ ( Vm ) 



, {ffly, + |m(m-l) y;} , > mi/ 10 + m (m- 1) v t v s . > 

 + ■ m* ^ 6 W"' + ^ ^ 8 W»> 



wv ]2 + m (w — 1) ty 8 + |m (m - 1) (m — 2) v* 

 m 6 



f»W)} (liv). 



We see that this expression converges much more rapidly than that for <£„ (r 2 ), 

 if m be at all large. 



The result (liv) might have been reached in a different manner. We might 

 have supposed the (ju,A) m_1 iVa individuals to have started from any element a 

 on the populated side of the boundary and taken mn flights without multiplying 

 to their final resting-place. The effect of this would be that a s = ^mnl 2 , and that 

 in the values of the v's we must write mn for n. But doing this gives us 

 precisely the coefficients of the i//'s in (liv). Thus (liv) is deduced directly 

 from (xlix). The proof becomes then much shorter, but it is more artificial ; 

 the fact that we may suppose all the unborn individuals to scatter from the 

 original centre is not so easily realised, and further it does not in the process 

 picture what takes place until the final arrangement after the with breeding cycle 

 is attained. In the method I have adopted we see the exact process of each 

 breeding multiplication, its increase of the operating factor by an additional Q t , 

 and its increase of the square of the standard deviation by an additional a 3 . 

 Lastly the final form (liv) enables us, without recalculating the v's for each 

 breeding cycle, to see very easily the effect in the case of any n-flight species, of 

 taking any number of breeding cycles. 



So long as we keep pA. constant of course our result for m breeding cycles 

 with n flights will be the same as for a simple scattering for mn flights of a 

 larger number of individuals. If /aA varies, however, we must adopt the method 

 indicated in the above proof, and work out each migration successively. The 

 same method must be adopted if a patch be rendered permanently sterile, because 



