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 



nlN= \ + \ ^^ e~4-^''(Za;' = i + TO„(«/(r) (xx); 



J n Jo V27r 



[Jn J-x/o-v27r J 



(VStt Jo v27r J 



= No- ]^=- m,(x/a)\ (xxi); 



W2ir ) 



. H/i./ = iVo-- 1 I +1 -^e ^'''' dx'\ 



(Jo J -x/<r V27r ) 



= N<j^[^ + vu{xjc7)\ (xxii). 



Now d = x + x, and 1'^ = /j,.,' — x'. 



' ■ '■ ' )) 



Hence - = ; -r-r-^ (x.xui), 



a I + m„ {x/a) 



y, {h + m., (x/a)} [^ + nio (x/<t)} - |-^ - m, (x/a)^ 



— = ; (xxiv), 



(T- [^ + mo{xla)\- 



say, for brevity. 



Here nii and mj are given by Table IX and | + nio is the ^ + ^a 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, O'l and 0-2, we have, from Tables II and IX, 



x'= 0, 1 + wio = -.500,0000, i + m., = -.500,0000, -^ - m, = -.398,9423, 



V27r 



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



x'=-2, „ =-579,2597, „ =-501,0512, „ =-391,0427. 



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



S7rf- = -5708, -5528 and -53-15. 



di 



