cxxii Tables for Statisticians and Biometricians [XXV 



Their differential equation will be 



1 dy _ 2mx' 

 y dx' CQ + a/ 2 ' 



leading to y = y Q (c + a' z ) m , 



2/3 2 19-5/8. 



where c = -5 ~ cH, and m = -^ . 



Pa o / pg o 



We can throw them into the following forms : 

 (i) a =l to 1-8 (mi =1 to 0), 



2 2/8, 2 , 19-5/8, 



where a? = _ - <r z , and wii = ~ -~ 3-- 



This symmetrical curve passes from two equal lumps through U -curves to a 

 rectangle. 



(ii) /3 2 = 1-8 to 3 (m, = to oo ), 



y * jfe ( i ~o?r > (ii)> 



2^ 2 5/3 2 - 9 



where a% = <r% and w 2 = 9 /o _ o \ 



This type of curve passes from a rectangle through limited range curves to 

 the normal curve (/3 2 = 3). 



(iii) fiz 3 to oo (w 3 = oo to |), 



v, 2 2 j 15/8 a -9 



where 3 2 = -= - = r, and rn 3 = 5 -= - . 



P2 ^> 2 p z o 



The limit p*-*- oo occurs when m 3 = | and O3 2 = 2o- 2 . This curve passes from the 

 normal curve through all grades of leptokurtosis. The limits of range in (i) are 

 from 01 to + !, in (ii) from a 2 to + 0%, and in (iii) from oo to + oo . 



We will now proceed to the probability integral of these three curves. 

 For (i) we have 



x' 



where B x (p, q) is the incomplete and B(p, q) the complete B-function, and 

 I* (p> q) their ratio. 



