66 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



become very large indeed. Each entry in column 5 of table 34 was 

 taken as the ordinate (y) at a given position (x) . As there was no appa- 

 rent biological reason for weighting differently the whorls in different 

 positions, all whorls were given equal weight. The method of fitting 

 was that of moments given by Pearson ( :02) . Since it is necessary in 

 this method to have an odd number of whorls, the first 29 were dealt 

 with instead of the whole 30. The origin was taken at a; = 15. The 

 observations were considered to give a system of trapezia, and the proper 

 corrective terms for the moments of such a system as given by Pearson 

 ( :02, p. 8) were used. The range is from 1 to 29, or we have 



21 = 28, or l = U. 

 For the quantities \ = -yf^, where z^', is the s^^ moment about the ori- 

 gin and I is half the total range, the following values were obtained, 



\ = 0.030132 K = 0.193246 



A, = .326446 \= .015303 



\,= .020219 \= .136507 



Further, we have 



Vo ^ 9.3454. 



From these values the following series of parabolas was obtained, 

 in which x is measured from the mid-range (= 15) . 



(I) y = 9.3454|l + 0.090397 ( j) \ 



(II) y = 9. 3454 |l. 025828 + 0.090397 (|) - 0.077483 (^ ) ^l 



(III) ?/ = 9.3454|l.025828+0. 034226 (j) —0.077483 (~) %0.093608;(j) | 



(IV) 2/=9. 3454 1 1.013268+0.034226 (^) +0.048110 (|) % 0.093608 (j) ^-0.146524 (|) H 



^v) 2/=9.3454|l.013268+0.036131 {j ) + 0.048110 ( j) Vo.084725 (|) ^-0.146254 (|) + 



0.008002 (^) ^1 



(VI) y = 9. 3454! 1.029148+0. 036131 (y) + 0.285372( |) V 0.084725 (|) % 



0.853919 [j) V 0.008002 (|) ^—0.733659 (|) 



In order to show the nature of the results with these parabolas fig. 

 11 bis has been prepared. This gives the observations and three 

 of the higher-order parabolas (namely, the third, fifth, and sixth) . It 



