ON THE- CONSTANTS OF INDEX-DISTRIBUTIONS AS 
DEDUCED FROM THE LIKE CONSTANTS FOR THE 
COMPONENTS OF THE RATIO, WITH SPECIAL REFER- 
ENCE TO THE OPSONIC INDEX. 
By KARL PEARSON, F.R.S. 
(1) Given x and y two variables, the frequency constants of both of which 
are known, we often require in statistics the frequency constants of their ratio or 
index : i = xjy. 
If the coefficients of variation are small, we have with the usual notation for 
means, standard deviations, moment coefficients, etc.*: 

+ ^^pj,z.p^: + -WziW + 2^,>„ ^ ) 
^ /|y ^ /.^: ^ ^ _ ^ ^ ^,1, 
\yj 2/ ^ y ^ y ) 
These formulae go to the order of the 4th power in the coefficients of variation, 
but of course this is not to the same order of approximation in M^, and M^. 
(2) It will be seen at once that these approximate formulae would be prac- 
tically unworkable if x and y were correlated, as we should have to find 3rd and 4th 
order product moments. 
* A rule denotes a mean value, o- a standard deviation, v, =(j-/mean, is a coefficient of variation ; 
/^2. /"3. are the moment coefficients for .r, fx^' , ^3', M4' for y, M^, il/3, for the index i, and 
Puv=S (a; - x)" (y - y)''IN, where N is the total number of pairs. Thus p3o = fj.3, 2^02 = /"2'> etc. 
