1895.] into the Measurable Characteristics of Plants, fyc. 371 



fraction destroyed as a consequence of this abnormality before reach- 

 ing maturity is 



It will, of course, be understood that little trust can be placed in 

 the absolute numerical results which are here put forward ; the point 

 which seems worthy of confidence, and which if it be indeed a reality 

 is of very great importance, is the form of the result. For by purely 

 statistical methods, without making any assumption as to the 

 functional importance of the frontal breadth, the time of life at 

 which natural selection must be assumed to act, if it acts at all, has 

 been determined, and the selective death-rate has been exhibited as a 

 function of the abnormality, while a numerical estimate which is at 

 least of the same order as the amount of the selective destruction has 

 been obtained. 



The method by which the result described has been arrived at is 

 likely to be capable of application to a very considerable number of 

 cases. Mathematically considered, the conditions which were at 

 first assumed and then proved to obtain in the organ discussed are by 

 no means general. It is necessary for the employment of this 

 method that the variations should be distributed on each side of the 

 mean with sensible symmetry, and that the position of minimum 

 selective destruction should be sensibly coincident with the mean of 

 the whole system. Such statistical information as is at present 

 available leads to the belief that these conditions may be expected to 

 hold for a large number of species, which are sensibly in equilibrium 

 with their present surroundings, so that their mean character is 

 sensibly the best, and the change of mean from generation to genera- 

 tion is at least very small. They cannot be expected to hold in cases 

 of rapid change such as those induced artificially by selection under 

 domestication, or naturally by rapid migration or other phenomena 

 resulting in a rapid change of environment. 



For the investigation of such rapid change, it would be necessary 

 to treat the more general case, in which the chances of deviations of 

 opposite sign are not sensibly symmetrical, and in which the mean is 

 not necessarily the position of minimum destruction. The treatment 

 of this case requires the help of a professional mathematician. 



It will be well to mention here a curious indirect confirmation of 

 the result just described, based on evidence derived from a quite 

 different source. 



An attempt has been made to show that physiological accidents of 

 a kind leading to change in the length of a portion of the carapace 

 affect a crab at a rate measured by the value of the quantity 1 e~^ 2 . 

 The symmetrical distribution of variations from the mean which has 

 been shown, especially by Mr. Galton, to occur in dimensions of 

 weight, length, muscular strength, and other characters of various 



