128 SHINKISHI HATAI. 



Stands is applicable to cases where the number of observations 

 is small. For cases where the number of observations is large 

 we must modify this still further. 



If we suppose the two limits to be mis d= d where s = in + n, 

 then equation (3) may be written in the following form : 



P = 



*Jm/s—9 



r ;t:™(l —xydx 



By successive integration by parts the denominator is evaluated, 



giving 



' m\n\ , . 



x'"(i—xfdx=- ^ — -^,. (4) 



X 



If we let X = injs -f s the numerator becomes 

 which is approximately 



— I £ 2mJi^3.^ 



s 

 With the above transformation the formula becomes 



P = 



{s + ly.m'^ff r+^ -^ 



in ! n 



I n\ 



e~^^'dz. (5) 



d 



Now if we apply Stirling's formula for large numbers (4) 

 becomes 



in ! 11 ! ;«"*;«" S ZTimn' 

 Therefore (5) may be written in the following form 



»+9 sSgS 



27Tmn J-9 



If we assume 



J 



2mn 



