inter alia, the percentage of the population in receipt of relief for every union in England and Wales on the 1st January of each census year, but there are irregularities. First, the return for 1850 is very defective, no information being available for many unions: secondly, the earlier return does not use as it should the figures taken from the "B returns" for 1st January, 1851, 1861, and 1871, but gives the percentage of paupers for 1st January, 1850, on the population for 1851, and so on. To keep this in mind I have labelled the plates, tables, &c., for these years 1850, 1860, and 1870; the last two returns are correct for 1881 and 1891. It scarcely seemed worth while to go through the considerable labour of forming correct returns for the earlier census years; for 1851, in fact, there are no published data. The returns given are sufficiently close to the decades to afford a clear picture of the advance or the reverse that has taken place in each successive period. The average pauperisms for the earlier years would, however, all be a little too low, owing to the populations being taken too high. The figures in these returns include paupers of all classes; lunatics and vagrants have not been struck out. It seems not unfair to exclude these two classes when comparing two individual unions, but in a case like the present, where we shall never mention any union by name but only consider the country as a whole, it appears better to retain them. The number of lunatics and vagrants in a union may not be due to any local conditions over which that union has control, but they form an important part of the pauperism of the kingdom, a part that there seems no reason to exclude when we are simply comparing the kingdom with itself at different stages of its progress. Owing to my having originally worked out the data for 1881 and 1891 not from the original returns, but from the figures given in the appendix to Mr. Charles Booth's work on the "Aged Poor," these data refer to registration districts and nct unions. In most cases the two are coterminous, but in a few instances two or more unions go to a district. There are now 647 unions in England and Wales, grouped into 632 registration districts; the constants, &c., of the distributions would be practically unaffected by such a small reduction by grouping. A frequency polygon showing the number of unions (in England and Wales) having any assigned rate of pauperism was formed for each of the years mentioned above, the unit of grouping chosen being o'5 per cent. (see the explanation of Table I below). The first four moment-coefficients of these five frequency polygons were then reckoned, and each of them was fitted first to a skew frequency-curve and then to a binomial-polygon, so as to condense the whole history into a few constants. The data referring to the statistical and to the theoretical distributions are given in the tables of the appendix as follows: Table I. The figures corresponding to the above five frequency polygons. Table II. Equations of the curves fitted to the polygons. Table III. Moment-coefficients; criterion; standard deviation. Table IV. Modes and means. Table V. Probable errors of skew-curve constants; range; skewness. Table VI. Constants of the point - binomials fitted to the polygons. This The plates contain the following diagrams:Plate I. Curve of mean pauperism (rates of total pauperism to total population) for each year from 1850-91. curve is based on the weekly returns, and gives the average for the year, not the figures for 1st January. It has been given for the sake of general comparison, and will not be much noticed in the sequel. Plates II-VI. The frequency-polygons, with the fitted curves and point-binomials superposed. Sec. 15. Table I is read thus: taking 1850 for example, there were in England and Wales : Only one union in which the proportion of the population in receipt of relief on 1st January lay between 0.25 and 0'75 per cent. Four in which the proportion lay between 0'75 and 125 per cent. Two in which the proportion lay between 125 and 175 per Thirty-three in which the proportion lay between 375 and 425 per cent. Forty-six in which the proportion lay between 425 and 4'75 per cent., and so on. The whole trend of the figures is shown by the black-lined polygons of Plates II-VI. Just glancing over these diagrams, we see at once that other changes, quite as important as changes in the average pauperism, have taken place. The standard deviation has decreased; the unions are more closely grouped round the mode. The period of stagnation, of positive reaction, from 1860-70 is characterised, as we see, not only by an increase in |