FIGS. 2 and 3.-Curves of frequency of all observed magnitudes of post-spinous lengths (fig. 2) and total carapace lengths (fig. 3) in 1000 adult female shrimps from Plymouth. The dotted curve is a probability curve of the calculated probable error: the continuous curve shows the results of observation. The horizontal scale represents thousandths of the body length: the vertical scale represents tens of individuals.

the value of r can be separately determined. This process has been performed for each of the values entered in the first two columns of Table I, and the results are recorded in the third and fourth columns.

In Table II, the results of a similar process are shown, but the individuals have been sorted into groups in each of which the length of the post-spinous portion of the carapace is constant, and the mean value of the total carapace length has been determined in each group. As before, the immediate results of these determinations are entered in the first two columns, and the deviation of each entry in these columns from its corresponding average, divided by its "probable error," is given in the third and fourth columns.

Table I.-Mean Value of Post-spinous Carapace Length (ps.) for every observed Value of Total Carapace Length (c) in Plymouth. (1000 individuals.)

The mean value of r, as determined from these two tables, is 0.81; that is to say, if all those individuals be chosen in which the deviation of one of the two organs from its average length is m times the probable error of that organ, the mean deviation from the average size of the other organ in the individuals chosen will be 0.81m times its probable error.

If the variations in size occurred in each of these two organs with a frequency equal to that indicated by a probability curve, then every pair of entries in each table should give the same value of r. As the conformity between the observed frequency and that indicated by probability is only approximate, the individual determinations give values which differ slightly from one another; but these variations are on the whole small, as may be shown in three ways.

In fig. 4, all the determinations of r are indicated: the deviation of the organ whose value is fixed in each case is measured parallel to the axis of y, the mean deviations of the associated organ parallel to the axis of x so that each pair of values in Tables I and II determines

Table II. Mean Value of Total Carapace Length (c) for every observed Length of Post-spinous Portion (ps) in Plymouth.

the position of one point in fig. 4. Those points whose position is determined from Table I, taking the carapace length as fixed, are indicated by crosses: those obtained from Table II, taking the postspinous portion as fixed, are shown by dots surrounded by circles O. The straight line drawn across the diagram indicates the ratio x = 081y; and every point, determined from either table, should therefore lie upon it. It will be seen that the points do, in fact, lie fairly closely round the line, though few lie actually upon it.

The accuracy of the assigned value of r may be tested in another way. When the value of r is known, it is evidently possible to calculate, from a known deviation of one organ, the mean associated deviation of the other; and in the fifth columns of Tables I and II the mean length of each dependent organ which should be associated with every observed value of the independent organ has been calculated on the assumption that r = 0.81. The length calculated in this way is found to agree very fairly well with the observed length, which is recorded in the second column of each table. The difference between the observed and calculated length of each dependent organ is shown in the sixth column.

A third way of checking the value of r is given by the ratio of the probable error of distribution of one organ to that of the other.

If

r0'81, then a deviation of carapace length equal to mQ. mm. is associated with a mean deviation of the post-spinous portion equal to 081mQps mm.; and from the values of Qe and Qps given above, the ratio 0.81 mQps: mQc is equal to 0.62. In the same way a deviation of mQps mm. in the post-spinous portion of the carapace is associated with a mean deviation of the whole carapace equal to 0.81mQc mm.; the ratio 0.81 mQc: mQps being equal to 1.05.

From the fundamental formula given above on p. 3,

Now = 0.590; and =

Q3ps
(3-5)2
Qe (4:55)2

= 0·591.

0.62
1.05

So that the assigned value of r fulfils all the required conditions very fairly well.

Having found a relation between the deviation of carapace lengths and that of post-spinous lengths, which is constant for all magnitudes of either organ in one local race, the question at once arises whether this relation is not a specific character of the shrimp, which is constant in all local races. At the beginning of the inquiry Mr. Galton suggested to me that the relation between the two organs indicated by the value of r was of such a kind that r might be expected to have the same value in all races of the same species, and in some cases in groups of species. A determination of the relation between carapace length and post-spinous length, and of other relations which are to be discussed below, has abundantly confirmed Mr. Galton's prediction.

In order to test the constancy of the relation between them, the variations in total length of carapace, and in length of the post-spinous portion, were measured in samples of adult female shrimps from Helder (North Holland), from Southport, from Sheerness, and from Roscoff (Finistère)-that is, from four places fairly distant from Plymouth, and differing from Plymouth and from each other in climatic conditions, in salinity and other characters of the sea water, and in nature of the sea bottom.* Each of these races was found to differ from the others in the average length of the organs measured, and in the probable error of distribution of each organ about its average; but the relation between the two organs, as measured by the value of r, was very fairly constant throughout.

The details of each determination of r are given in Tables III—X, at the end of the paper; they are constructed on precisely the same plan as that used for Tables I and II, and need not therefore be further explained. The values of r deduced from all the tables are―

The approach to identity between these values is very striking. The differences between them are certainly large; but they are not, as it seems to me, larger than the probable error of each determination. The number of individuals employed, even in the races from Plymouth

* I am glad to express my gratitude to Dr. P. P. C. Hoek, to Professor Delage, and to Messrs. W. Garstang and W. H. Shrubsole, for their kindness in procuring for me these samples.