116 Proceedings of Royal Society of Edinburgh. [sess. 
On the Distribution of the Proper Fractions. By 
Duncan M. Y. Sommerville, D.Sc. Communicated by 
Professor Chrystal. 
(MS. received 15th January 1906. Read March 19, 1906.) 
CONTENTS. 
Sec. Page 
Introduction . . .116 
1. Normal Distribution . . 117 
2. General Theorems on the 
Normal Distribution . 119 
3. Method of writing down 
N.D 120 
Tables. N.D. for */4>12 
and */> 13 . . .121 
4. Other “ even ” Distributions 122 
5. Evenest Distributions by 
the Coalescing of Pairs of 
Classes . . . .124 
6. Equations satisfied by the 
Weights of the Loaded 
Denominators . .124 
Sec. Page 
N. D. is an Even Distribu- 
tion for Symmetrical 
Loading . . .125 
7. Division into more than n 
Classes . . . .125 
8. Division into fewer than n 
Classes . . . .125 
Submultiple Division . 126- 
9. Method of writing down 
any distribution . . 125 
Table. Distribution of 
*'/ :J> 24 into 8 Classes . 127 
10. General Equations for any 
Distribution . . .127 
11, 12. Effect of omitting the 
Lower Denominators . 128 
In statistical work which deals with integral variates, the data 
frequently appear in the form of ratios, or unreduced proper 
fractions, e.g. sex- and fecundity-ratios ; * and to facilitate com- 
parison these are arranged in classes, all the ratios falling within 
the same class being considered as equivalent. These classes, 
must, as far as possible, contain an equal number of the ratios. 
Further, in certain fields the different kinds of ratios do not all 
occur with the same frequency, that is, one denominator will 
occur more frequently than another, without any reference to the 
number of fractions having this denominator. In such cases the 
fractions must be distributed in such a way that, if to each 
fraction having a particular denominator p there be assigned a 
multiplier or weight fx p , in each class the sum of the numbers of 
* Cf. Karl Pearson, “ On the Inheritance of Fecundity in Thoroughbred 
Brood-mares,” Phil. Trans. A, vol. cxcii. pp. 294-296 (1898), where the 
evenness of various distributions of fecundity-ratios is discussed empirically. 
For this reference, and also for suggesting the problem, I am indebted to 
Mr David Heron, M. A., who is at present studying under Professor Pearson. 
