Robertson, Further Explanatory Remarks Concerning etc. 31 



Moeser thinks that this equation is superior to my equation (5) 

 because, he believes, it avoids the assumption that ( daily 



increment) is a maximum when growth is half completed (x l / 2 A). 

 In this belief he is mistaken, however, as the following considera- 

 tions show: 



Differentiating equation (7) we obtain: 



(8) 



which is identical with equation (1). 

 Differentiating again we obtain: 



. . (9) 



J2 



hence - -^ is zero when x V 2 A ? an( ^ f r tnat an ^ all values 



Q I 



Q X 



s negative, hence-, is a maximum when x = 1 / 2 A 4 )- 



etc 



Consequently Moeser's equation involves, just as much as 

 mine does, the assumption that the rate of growth is a maximum 

 when the growth-cycle is half completed. Now it is to be recol- 

 lected that, especially when we rely upon observations upon the 

 growth of a single individual, any single measurement may chance 

 to be erroneous, and if we incorporate into our equation such a 

 measurement our equation will also be erroneous. In employing 

 my equation it is possible to eliminate this source of error to the 

 greatest possible extent, for K and t x are determined, not from 

 any single observation but from all of the observations by the 

 method of least squares. By employing this method the -f- errors 

 attaching to certain observations are cancelled by the errors 

 attaching to others and the constants thus computed, as all physicists 

 know, are much more nearly ideally correct that constants computed 

 from single observations. In employing Moeser's equation, how- 

 ever, we are forced to rely absolutely upon the accuracy of a single 

 observation, namely, the measurement of the length, volume or 

 weight w r hen t = o, and then, if this observation should chance to 

 be erroneous (owing to the intrusion of adventitious variables such 

 as fluctuations of temperature etc.) the whole equation will share 

 in the error and the experimental results may not fit the equation 

 at all owing to a single experimental error. 



Moeser states that ,,Die Roberts on'sche Interpretation der 

 autokatalytischen Formel ist direkt falsch. Daher ist es nicht 

 wunderbar, dass seine Zuwachswerte rnanchmal urn die Halfte von 



4) Cf. J. Todhunter Differential Culculus" 7th Edn. London 1875, Chapter 13. 



