r = 0.81, 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 0.81mQ. mm.; the ratio 0.81 mQe: mQps being equal to 1.05. From the fundamental formula given above on p. 3, 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. and Southport, is too small to allow of a satisfactory determination of the second decimal place. The reader, who cares to do so, may satisfy himself of this by taking 0.80 or 0.82 as values of r in the Plymouth tables, when he will find the agreement between calculated and observed values of the associated organ to be only slightly less close than in the existing table. It may, therefore, be fairly said that the values of r obtained by examining five races of shrimps are not inconsistent with the existence of a constant value in all the races examined-a value lying somewhere between 0.80 and 0.85. So that if the deviation of total carapace length from its average be expressed in terms of its probable error, and if the deviation of the post-spinous portion be in the same way expressed in terms of its probable error, then, when either organ differs from its average by any constant amount, the mean deviation of the other will be a constant fraction of that amount, the fraction being between 0.80 and 0.85. A similar approximation to constancy has been found to exist in the relation between three other pairs of organs, determined in the samples from Plymouth and from Southport. These organs are so largely independent that the probable error of the determination is much greater than before; and accordingly the irregularities in the results are much greater than in the previous case. The mean values of r are as follows: These values are found from the tables at the end of the paper, from which it will be evident that the determination is much less reliable than that first discussed. So far, we have only investigated the mean deviation of each organ which is associated with a known constant deviation of another organ. But since a fixed deviation of one organ does not as a rule involve a fixed deviation of the second, it becomes necessary to inquire how the values of this second organ are distributed about their mean. Mr. Galton points out that the deviations of each dependent organ are distributed about their mean with a probable error of Q√1-r2. So that in Plymouth, for example, those post-spinous carapace lengths which are associated with any fixed total carapace length should be distributed about their mean, with a probable error of Qps 1-(0.81)* or 3.5 034, or 2.03 nearly. In the same way, the values of total carapace length which are associated with a fixed length of the postspinous portion should be distributed about their mean with a probable error of 4.55 0.34, or nearly 2.64. Therefore, when the Plymouth shrimps were sorted into groups, such as those used in Table I, such that the carapace length was constant in each, then the post-spinous lengths in each group should have been distributed about the mean given in the table with a probable error of 2:03. The largest of these groups contained only about eighty individuals, so that any accurate determination of this point was out of the question; but a rough estimate of the probable error was made in each group which contained more than forty individuals, and the mean value obtained in this way was 1.96, which is perhaps sufficiently near to 2:03. A similar treatment of the groups in Table II gave 2:59 as the probable error of distribution; and this, again, is not very different from 2·64. The other samples show a similar rough agreement between the probable error of the dependent organ in each group and that indicated by the value Q1-; but the numbers employed are too small to make a determination of this point worth serious discussion. It cannot be pretended that the results here given are either sufficiently numerous or sufficiently accurate to serve as a basis for generalisation; but at the same time they seem certainly to suggest a very important conclusion. For if the values of r have really the degree of constancy which has been attributed to them, then by expressing the deviation of every organ examined from its average in terms of the probable error of that organ, the deviation of any one of these organs from its average can be shown to have a definite ratio to the associated deviation of each of the others, which is constant for all the races examined. And since both the organs measured and the samples of shrimps examined were chosen, in the first instance, by chance, any result which holds for all these organs through all these races may be reasonably expected to prove generally true of all organs through the whole species. That is, the results recorded lead to the hope that, by expressing the deviation of every organ from its average in Mr. Galton's system of units, a series of constants may be determined for any species of animal which will give a numerical measure of the average condition of any number of organs which is associated with a known condition of any one of them. A large series of such specific constants would give an altogether new kind of knowledge of the physiological connexion between the various organs of animals; while a study of those relations which remain constant through large groups of species would give an idea, attainable at present in no other way, of the functional correlations between various organs which have led to the establishment of the great sub-divisions of the animal kingdom. |