180 
Miscellanea 
each of them is counted Writing the classes horizontally with the fractions *//> in columns 
under their respective denominators, the n.d. for n = l2 is represented as follows: 
0 0 0 0 0 
T 1 
T T 
I 2 
T 2 
T 2 
2 3 
2 3 
3 4 
1 2 3 4 5 
9 10 11 
12 
0 
0 
0 
0 
0 
0 
0 
1 
T 
1 
1 
1 
1 
1 
i„ 
T 
2 
2 
2 
2 
3 
2 
2 
3 
3 
3 
3_ 
4 
2 
3 
3 
4 
4 
4_ 
5 
3 
3 
4 
4 
5 
5 
fl"_ 
6 
8 9 
9 10 
10 11 
9 
f 
10 
A bar denotes that the fraction is counted g. 
II. Giving weights fi^ to the various denominators and expressing that the normal dis- 
tribution is even, we get a series of equations, 
i.e. the frequency-curve for the denominators must be symmetrical. 
If we divide the fractions */5>?i into w-|-hi classes, then we have to divide the fractions 
*/^{n + m) normally and consider n„=0 if p>n and therefore also if p<m. Also if p>m, 
A'p = M)i + »rt-p- Hence if the frequency-curve is symmetrical from m to n («-|-»« being even), and 
if the denominators <m may be neglected, we may divide the fractions *jlf7i into n-\-m classes, 
and then make pairs of classes from the second onwards coalesce. We have then the distribution 
11 3 
into ^ (•« -t- m-\-2) classes, 0 to — — , — : — to 
n-^in n + m n+vi 
p=n 
., each of weight 2 ^p. 
The general case where the frequency -curve is skew is not solved, but by considering it as 
symmetrical about the mode, e.g., and neglecting the lower denominators, as above, it may 
be possible to obtain an approximate solution, provided the mode lies among the higher 
denominators. 
For other results which are only true when the extreme fractions are counted ^ we may 
refer to the complete paper. 
The following graphical method of obtaining the first theorem was communicated to me by 
Professor Steggall. 
Any positive fraction can be represented uniquely in the jiositive region of the plane by 
a point whose coordinates x, y are respectively the numerator and denominator. The lines 
i(/ = 0, x=y^ x = a will then confine all the proper fractions These are divided into the 
classes described above by the lines rx^ny (/• = 0, 1, «). The number of fractions in the class 
