no marked tendency either to rise or fall. With these 1923 individuals a curve of frequency was drawn (fig. 2). Its constants were calculated by the method employed by Professor Karl Pearson (‘Phil. Trans.,’ vol. 186). Range of variation = 1227—1370 thousandths of standard, unit of deviation is 0*004 of carapace length ; therefore the observed range = 36 units, reckoning from 1227 upwards.
Centroid vertical (= position of arithmetic mean) = 19T02964. The second moment about the centroid ( 2) = 26*400476. Standard deviation (error of mean square), a — ^ = 5T38139. Third moment (/19) = 0*681766. Fourth moment (^4) = 2203*762099. A> which is n3 ^/fi^ — 0*000025 ; /?2 = .v-i/fi* = 3*161849. The critical function 2/32—3y3i—6 = 0*323623 is positive, and so the theoretical curve has an unlimited range.
Professor Pearson’s measure of skewness for a curve of unlimited range is given by the formula
2 ^ A r — 2 r + 2 where r = £ i 2 / J . - 3 A - 6 Here r = 40*080414 and skewness = 0*002262. It is clear from the values of the constants that the generalised probability curve would not differ perceptibly from the symmetrical normal curve, where A = 0 and /32 = 3.
The areal deviation of the curve of observation from the normal curve is only 5*1 per cent, of the whole area.
Frontal Breadth.—It will be seen from the table that throughout the life of the crab the mean of this dimension falls steadily; as the crab grows the forehead becomes relatively shorter. On this account it is difficult to obtain a satisfactory idea of the distribution of deviations. The means of groups 6—7 do not differ widely, and so with these the constants of variation were calculated. The observed range throughout the whole series was 640—795 thousandths of standard, and so there are o9 of our units of deviation. The range in groups 6—7 (including 460 crabs) was 648—747 thousandths, that is, 25 units.
Centroid = 13*791305. = 11*236156. a = 3*352036. = 2*800794. 114. = 442*572048. A = 0*005529. A = 3*505490. r 15*084346. Skewness = 0*02845.
Here again the critical function 2/32—3/3i — 6 — 0*994393 is positive
