POSITION REGRESSION— PRIMARY BRANCHES. 57 



size of the whorl is a function of its position on the axis? So far we 

 have demonstrated positional differentiation of broad classes of whorls. 

 It now remains to determine whether this extends to the individual 

 whorls, and in general to find out the laws of growth which result in 

 main-stem or primary-branch, or secondary-branch, or other whorls, 

 showing the particular characteristics in their variation which they do. 

 Cases of positional differentiation in like parts are well known in a 

 number of plants, ' but a complete analysis of the phenomenon which 

 takes into account the whole series of repeated parts or characters for 

 a large and richly branching plant like Ceratophyllum has not hitherto 

 been undertaken. 



THE RELATION BETWEEN THE NUMBER OF LEAVES IN THE WHORL 

 AND POSITION ON THE PLANT. 



We come now to the more direct investigation of the morphogenetic 

 laws concerned in the formation of the leaf -whorls. From the method of 

 growth in lateral branches of Ceratophyllum it is clear that we have 

 an almost ideal form for such a study. The whorls on a branch present 

 a linear series of parts in which we know the order of formation in time 

 (cf. p. 10, supra). Such a system suggests at once a number of very 

 interesting problems in morphogenesis. It will conduce to clearness, 

 if we base the discussion of our results on such individual problems, 

 taking up the different ones in order. 



POSITION REGRESSION IN DIFFERENT PORTIONS OF THE PLANT— THE 

 FIRST LAW OF GROWTH IN CERATOPHYLLUM. 



The first problem which logically presents itself may be stated in 

 this way: In a series of like parts produced in a regular ordinal succes- 

 sion what relation exists between the form of a particular member of 

 the series and its ordinal position? If there is a change in the character 

 of successively formed parts, what law governs this change? It is 

 obvious that this problem is a very fundamental one, because of the fact 

 that one of the most frequently occurring plans of structure which we 

 know in the organic world is the metameric. In this type of structure 

 the organism is built up of a series of primitively similar units arranged 

 in linear order. Our problem is to find out, if possible, in a very simple 

 case the laws of differentiation in such a system. The biometrical solu- 



^Many examples are given by Miss Tammes (:03) and the subject has been inves- 

 tigated biometrically by Pearson ( :05) in Asperula odorata and Equisetum arvense 

 and by Pearson and Radford ( :04) in beech leaves. 



