294 On, Generalised Tchehycheff Theorems 
Now a good approximation to q.^ by (xiii) must be ^ (/Sj + /^a') + (1 - ^'') i 
hence substituting 
= {{^2 - 3) (cos^ d + r= sin'' ^) + (^S^ - 3) (sin* ^ + cos^ ^) 
+ 3 -4(1 -r^)cos^(9sin=^j 
For the special case of normal distribution, if we write «^ = 4 (1 — r^) cos^ 6 sin" 6, 
Again 
and for a normal distribution, 
^ (15-9/c=) 
l6 
Further general cases can be at once written down, but it will suffice to give 
hei-e the leading values of Ig for a normal distribution : 
1 
I. 
le 
_ 1 
R'' 
h 
1 
R'' 
~ A," 
h 
_ 1 
R'^ 
R^' X 
,, = rj„(945-1050/c^+ 225/c*), 
The following table gives the maximum Tchebycheff limit for the probability of 
an individual falling within the circle X Vo-i^ + o-j^ for various values of 
= 4 (1 - r=) o-,V2V(o-/ + 0-/)^ 
(Is) denotes the particular / from which the maximum limit is found. (Ig ?) 
denotes that the corresponding numerical value is a Tchebycheff limit found from 
/s, but it is not known whether /„ would not give a higher value, Ig not having 
been tabled. The second part of the table provides the values of 7s from which 
the first part has been computed. They may be useful in the determination of the 
Tchebycheff limits for other values of \. 
