POSITION REGRESSION— PRIMARY BRANCHES. 69 



ordinate was expressed as a deviation (in excess or defect) from 7. 

 The origin of x was taken at 0, i.e., at the proximal end of the branch. 

 A curve of the type 



1) = A-\r C (log X) 



was then fitted by the method of least squares. 



It gave on the whole a very good graduation; so good as to indicate 

 clearly that one was on the right track. It signally failed, however, to 

 give a good result at the lower end of the range. It bent altogether too 

 gradually at the start to represent the facts. Weighting the ordinates 

 with their observed frequencies did not help matters at all. While in this 

 way a better fit at the start of the range was obtained, it was at the 

 expense of getting very improbable values for whorls beyond the 10th or 

 12th. The results clearly showed, however, that some logarithmic curve 

 offered the solution of the problem. It only remained to find the proper 

 logarithmic curve. 



The next assumption to be tested was that the true law of growth 

 was such that 



dy \ . 



-r~ = const. 



ax x—o. 



or, in other words, that the rate of increase in mean leaf-number varied 

 inversely as the position measured from a fixed point (a) on the axis. 

 This leads to a curve of the form 



y = A + C log (x — o). 



In fitting this curve a value of 0.8 for the constant « was first found by 

 a rough method of approximation. Then putting this value of 0.8 for a 

 into the equation, the constants A and C were found by the method of 

 least squares. This gave a first approximation to the curve. The next 

 step was to proceed to get a better fit by modifying all the constants by 

 small amounts, the modifying terms being calculated by the method of 

 least squares. The final equation determined was 



y = 0.9520 + 1.3608 log (x - 0.8015) 



Remembering that y has been measured from 7, we have for the 

 final result 



Y = 7.9520 + 1.3608 log {x - 0.8015) .... I 



where Y denotes the actual mean number of leaves per whorl and x the 

 position of the whorl. Calculating the ordinates of this curve corre- 

 sponding to the successive ordinal positions of the whorls, we have the 

 series of values given in the second column of table 35, against which 

 are put for comparison the actually observed values. 



