IX — X] Introduction xxvii 



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

 truncated portion, and N be whole population. Then 



njN= f + f -j== e'^dx' = \ + m {xj<r) (xx); 



Jo Jo v27r 



(Jo J -xi**j2ir J 



(\/2tt Jo v2tt J 



= N, 



a 



==- m,(a;/o-)| (xxi); 



nu' = N*-\\ +| =6-**"^ 



roo rO x 'i 



1 + 1^ 



I Jo J -i!a N'ltr 



= TVcr 2 {£ + TO, (a'/o-)} (xxii). 



Now d = x + x, and S 2 = /i/ — a?. 



1 a; 



j ~7S=* ~ ^ W°") + - (i + w (x/o-)} 

 „ c* v2tt <^ , .... 



Hence — = ■ , , . (xxm), 



a i + m (x a) 



.(xxiv), 



{J + m, (a/or)) {4 + to (a?/o-)} - j-^== - m, (ar/<r) 

 ^ = l4 + 7»,(«/a)}» 



£2 ( i + m2 )(« +mo )_(^=- Wl )' 



and d? = ~T~i F~ (xxv) ' 



say, for brevity. 



Here m x and m? are given by Table IX and \ + m„ is the £ + i<* 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 x' = 0, Ol and 0"2, we have, from Tables II and IX, 

 x'= 0, 4 + w„ = -500,0000, \ + to 2 = -500,0000, — L -to 1 = -398,9423, 



V27T 



x'=-l, „ =-539,8278, „ =-500,1325, „ =-396,9526, 



x'=% „ =-579,2597, „ =-501,0512, „ =-391,0427. 



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



2 2 /d 2 =-5708, -5528 and -534.5. 



di 



