226 Mr. E. Warren. 



no marked tendency either to rise or fall. With these 1923 indi- 

 viduals a cnrve 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 = 12271370 thou- 

 sandths 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) = ] 9*102964. 



The second moment about the centroid (/tg) = 26*400476. 



Standard deviation (error of mean square), a = ^ ' /JL Z = 5*138139. 



Third moment (^) = 0*681766. 



Fourth moment (^ 4 ) = 2203*762099. 



ft, which is rfltf = 0-000025 ; ft = mltf = 3*161849. 



The critical function 2ft 3ft 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 



here r = 



2ft-3ft-6 



Here r = 40*080434 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 

 ft = and ft = 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 devia- 

 tions. 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 39 of our units of deviation. The range in groups 

 6 7 (including 460 crabs) was 648 747 thousandths, that is, 2& 

 units. 



Centroid = 13*791305. ^ = 11*236156. 



<r = 3-352036. K = 2*800794. 



to = 442*572048. 

 ft = 0*005529. ft = 3*505490. 



r = 15*084346. Skewness = 0*02845. 



He re again the critical function 2ft 3ft 6 = 0*994393 is positive 



