20 C. V. L. Charlier. 



TABLE v. Farameters of frequency curves for pure lines of Phaseolus vulgaris. 





I-'-,) 



h 



c 



0 



ß4 



Yii - Mean 



G 















■\veiglit 



in m g 



a ; m 



Line A 



145 



+ 3.338 



2.177 



+ 0.039 



— 0.022 



642 



109 



0.170 



' „ B 



475 



4-1 .058 



1.851 



4-0.032 



— 0.001 



558 



93 



0.166 



„ C 



282 



4-1.585 



1.524 



— 0.023 



— 0.015 



554 



76 



0.138 



„ D 



307 



+ 1.453 



1.681 



+ 0.120 



+ 0.031 



548 



84 



0.156 



„ E 



255 



4-0.737 



1.497 



— 0.069 



4-0.021 



512 



75 



0.146 



T 



141 



-1-0.624 



1.276 



+ 0.036 



— 0.017 



506 



64 



0.126 



„ Q 



305 



-1-0.341 



1.368 



4-0.013 



+ 0.028 



492 



68 



0.139 



„ « 



159 



-1-0.277 



1.427 



+ 0.113 



4-0.038 



489 



71 



0.146 





241 



4-0.137 



1.517 



— 0.010 



4 0.004 



482 



76 



0.158 



„ 



533 



— 0.200 



1.562 



— 0.031 



4-0.082 



465 



78 



0.167 



-, H 



418 



— 0.395 



1.396 



+ 0.033 



0.005 



455 



70 



0.154 



,, R 



83 



— 0.398 



1.308 



4 0.094 



± 0.000 



455 



65 



0.144 



„ ■^ 



712 



— 0.411 



1.477 



4 0.045 



± 0.000 



454 



74 



0.163 



„ P 



106 



— 0.443 



1.493 



— 0.094 



+ 0.044 



453 



75 



0.165 



V K 



188 



— 0.511 



1.323 



+ 0.002 



4 0.026 



449 



66 



0.147 



„ L 



273 



— 0.576 



1.383 



— 0.007 



4 0.026 



446 



69 



0.155 



„ M 



295 



— 0.932 



1.432 



+ 0.048 



4-0.004 



428 



72 



0.168 



„ N 



357 



— 1.344 



1.560 



-1-0.094 



4 0.032 



408 



78 



0.191 



„ 0 



219 



— 2.474 



1.299 



4-0.166 



+ 0.075 



351 



65 



0.182 



The lines are here ordered according to their mean weight, which varies between 

 642 mg and 351 mg. The standard deviation (^j varies between 109 mg and 64 mg 

 and seems to depend on the magnitude of the beans, being nearly proportional to 

 their mean weight. This fact is shown by the last column, which gives the quotient 

 between the standard deviation and the mean weight of the beans. Taking the 

 mean of the numbers in the last column, we find that the standard deviation 

 amounts to 15,7 7o of the weight of the beans. 



The frequency curves of most pure lines show a good agreement with the 

 normal curve. Some pure lines, however, have a frequency curve with a notable 

 skewness, as the lines D, S, R, P, N and 0. The greatest value of the skewness 

 occurs for the line 0, where ßg = + 0.166. As to the excess, we find that a nega- 

 tive excess occurs rather seldom. The greatest positive excess occurs at the lines G 

 and 0, amounting at the most (in the line G) to + 0.082. The form of the fre- 

 quency curve of the line 0, which has the greatest deviation from the normal 

 curve, is shown in fig. 10. In fig. 8 and 9 I give the frequency curves of 

 the lines A and G. The agreement between theory and observation is generally 

 tolerably good, the most notable exception occurring in line 0, where the number 

 of beans with the mean weight 375 mg seems to be too small. 



On the connexion between the' values of the parameters and hereditary cir- 

 cumstances I have made some researches, till now with negative result. When it 

 becomes possible to compare the results from many generations, it seems probable 

 that such a study will show itself more fertile. For the present I will only point 

 out the simple and instructive description of a frequency curve that is given through 

 the coefficients ßg and ß^. They give a most concentrated idea of the curve and 

 allow one to calculate the theoretical frequency curve in the most simple manner. 



