The dimensions chosen were:- -(1) the "frontal breadth "-the distance in a straight line between the tips of the extra-orbital teeth of the carapace (from the point A, fig. 1, to the corresponding point on the opposite side); and (2) the "right dentary margin," measured in a straight line from the apex of the first to that of the last lateral tooth (from A to B, fig. 1). The "length" of each crab was taken as the length of the carapace, from the tip of the middle interorbital tooth to the posterior margin (from C to D, fig. 1).

This is, of course, not the total length of the body; but the curvature and flexibility of the abdomen render an exact determination of the real body length very difficult.

In order to compare the variability of a dimension in crabs whose carapace is only 7 mm. long with that of the corresponding dimension in adult crabs, whose carapace length is from 40-50 mm. or more, it is evidently necessary to adopt some method of picturing the crabs as of one standard size; and accordingly the measures obtained have always been expressed in terms of the carapace-length of the crab to which they belong, taken as 1000. The measurements were made by means of a screw, of 1 mm. pitch, carrying the object across

the field of a microscope, and by means of graduations on the head of the screw the observations were recorded to the nearest hundredth of a millimetre. It is believed that the probable error of any observation is not much more than one hundredth of a millimetre. In order to minimise the effect of errors of observation, the results, after being expressed as fractions of the carapace-length, were sorted into groups, such that the measures in each group did not differ by more than 0.004 of the carapace-length, and all measures in the same group were treated as identical. The unit employed in tabulating the results was therefore 0.004 of the carapace-length; but in what follows the results are expressed, for the greater convenience of the reader, in thousandths of the carapace-length. It will be noticed that the principal effect of this alteration upon the results is to diminish their apparent regularity-an aberration of one unit of measurement appearing as four units in the tables below.

1. Variation in Frontal Breadth.

An initial difficulty in determining the error of distribution of frontal breadths about their mean in young crabs, arises from the great rapidity with which the mean itself changes during growth. The mean frontal breadth in the smallest specimens was found to be 853 14 thousandths of the carapace-length, while at maturity it is only 604-94 thousandths. The rate at which this change occurs can be gathered from the following table of the crabs measured, and the same result is graphically shown in fig. 2

the

From this table it appears that the mean frontal breadth changes at such a rate that when the carapace-length has increased 0 2 mm., frontal breadth has almost always diminished by less than four thousandths, that is to say, by less than one of the units of measurement here employed. For the purpose of the present investigation the mean was therefore considered stationary during every period of increase in size of not more than 0.2 mm., and the young crabs were accordingly sorted into groups, the individuals of each group differing by less than 0 2 mm. in respect of their carapace-length. The distribution of frontal breadths about the mean was then examined in each group separately.

As the difference in size between the largest and the smallest of the growing crabs was 7 mm., it follows that the material was divided into thirty-five groups. This subdivision of the material had great disadvantages, because, instead of a single group of over 7000 individuals, varying about the same mean, from which a fairly reliable indication of the law governing frequency of deviation might have been expected, the average number of individuals in any one of the available groups was only 200; and from so small a number of obser

Table I.-Mean Frontal Breadth (F) expressed in thousandths of the Carapace-length, corresponding to various Carapace-lengths (C), together with the Number of Individuals on which each Determination is based.

[Note. The carapace-length given in the table is the mean of all lengths included in each group. For example, the entry 71 includes all crabs measured in which the carapace-length was between 700 mm. and 7·19 mm., and so on.]

The

vations no satisfactory demonstration of the law of variation at any given moment of growth could be obtained. Nevertheless it was necessary, before proceeding further, to ascertain with some certainty what the law of variation through the whole series really was. belief in which the work was undertaken was, that the law of variation would be found throughout to be that of the ordinary probability equation; and this belief was tested in the following way:-In each of the thirty-five groups, the arithmetic mean of the frontal breadths, and the mean of all the deviations from it, were determined; and from the " mean error" found in this way the modulus of the probability function was calculated. Then, by calling the mean of each group zero, and expressing the deviations from the mean in terms of the modulus, a number of curves were obtained, in each of which the modulus was unity and the mean zero; a similar curve of adults was constructed, and the corresponding ordinates of all the thirty-six curves so obtained were added together. It is evident that, if the chance function really expresses the law of variation throughout the series, then the curve resulting from the treatment described will be a symmetrical probability curve of unit modulus. The actual

crepancies, the general agreement between the observed frequency of tables of the probability integral in Table II. In spite of some disof every observed deviation is compared with that indicated by the curve obtained is plotted in Fig. 3, and the frequency of occurrence

FIG. 2.-Diagram to illustrate the Change in the Mean Value of the Frontal Breadth with Growth in Carapace-length.
Ordinates represent fractions of the Carapace-length. Abscissa represent Carapace-length in millimetres.

FIG. 3.-Distribution of Frontal Breadths in 8069 Female Crabs from Plymouth Sound, old and young. Deviations expressed in terms of the Modulus. The three cases of deviation greater than three times the Modulus are omitted.

deviations and that indicated by the probability integral is fairly close. The mean error of the observed curve is 0.5621, whereas it should be 1/= 0.5642, the difference between the two figures being less than 0.5 per cent. The error of mean square is 0.7123, instead of 0.7071, a difference of less than 1 per cent. The sum of the squares of the positive deviations is 2115. The sum of the negative deviations is 1992. The total number of individuals of deviation more than 01 is 3593 on the positive, 3574 on the negative side, a difference of about one-half per cent.

On the whole it may be said that the result agrees with that given by the theory of probability as well as could be expected from the number of observations, and that the law of frequency of variation throughout the series may, as was hoped, be assumed to agree with the ordinary law of chance.

From the result so far obtained it followed that a determination of the quartile deviation, or any other of the constants of the pro