1905 - 6 .] Distribution of the Proper Fractions. 
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This may be written 
i(n— 1) or 
or J(w-2) 
1 1 
[If n is even p in will not enter, for ei n = e' in = J.] 
{( e p ^ p)pp "f" { e n—p f n—p)Pn—p} ^ • 
Now 
(§ 2 } 
and the equation becomes 
\(n— 1) or l(n— 2) 
^j( € P e p) (ftp Pn—p) 
1 
Now, by taking the classes in pairs, since the distribution is- 
symmetrical about the middle class or classes, we shall obtain 
J (' n — 1) or \{n — 2) distinct equations of this form, according as n 
is odd or even. But this is the number of the quantities^ 
/Xp - fx n _ p , and since the equations are homogeneous and linear 
in these quantities they can only be satisfied by the quantities- 
f j.p - [x n _ p all vanishing, i.e. 
for all values of p from 1 to n — 1 .* 
Hence the normal distribution will be an even distribution also 
in the case where the frequency curve for the denominators is 
symmetrical. 
§ 7. Suppose now that we divide the fractions */^\>n into more 
classes than n, say into n + m, and consider the possibility of the 
sums of all the classes being equal ; it is the same thing as if we 
divide the fractions */ Ij> (n + m) normally and consider p p = 0 if 
p>n , and therefore also if p<m. Also if pzfm, y p = y n+rn _ p . 
§ 8. Next, suppose the fractions */^>w to be divided into fewer 
classes than n, say into m, and let n = xm + y, where y<m. 
* The only alternative is the vanishing of the discriminant of the system 
of equations. It is easily seen that this disct. is the determinant which 
corresponds to the left-hand upper quadrant of the normal distribution, 
omitting the first two rows and the middle and left-hand columns, in such a 
way that the elements corresponding to fractions occurring in only one class- 
are 1, those corresponding to the limits and the other elements 0. 
Whether the disct. vanishes or not, the equations are always satisfied by the- 
above values. 
