IX X] 



Introduction 



xxv 



Let x be the distance from stump to centre of curve, n equal the area of 

 truncated portion, and N be whole population. Then 



n/N=T + I 1 '' ^=e-* x '*dx' = $ + m (x/<r) ............... (xx); 



J JO V27T 



I r<*> ro nJ i :) 'l 

 nx^NaU + -f=e-**daf\, 

 (Jo }-&/%* ) 



( 1 f" X' -il'2 , ,) 



= Na \ -=e dx'\ , 



(V27T JO V27T ) 



= No- \-== - TO, (/)[ ................................. (xxi); 



IV27T 



, v ...^ir + r ..-*[ 



I Jo J-zlvVZTr 



{fz/<r T 'i , ,, 1 



i+f -J= e -^ 2 cfa' 

 Jo V27T j 



= ^<7 S {i + wi, (^/tr)} ....................................... (xxii). 



Now d = x + x, and 2 2 = /*/ S*. 



1 a; 



rf \/i^ ~ "^ ^*^ + ff ^ + m ^*^ 



Hence -= - - - - ........................... (xxiii), 



a- $ + fo (xja) 



s , {i + m, (*/<r)) (i + m (*/)) - J^L - m, (*/r)J 



^ = {i + ^^M}* ...... (XX1V)> 



2 , g +OTt) (H>.)--^ 



and = 



say, for brevity. 



Here m, and m,, are given by Table IX and J + m is the + J^x of Table II. 



Formula (xxv) has not yet been tabled for different values of x, as it occurs 

 much more rarely than the corresponding function for a true tail. 



If we take three values a! = 0, O'l and 0'2, we have, from Tables II and IX, 



x = 0, i + m, = -500,0000, \ + m, = -500,0000, 



'=!, =-539,8278, ='500,1325, 

 x ='2, =-579,2597, ='501,0512, 

 Whence from formula (xxv) for the three values of x 



'5528 and 



V2?r 



, = -398,9423, 



='396,9526, 

 ='391,0427. 



