MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 401 
= cs x’, approximately.* 
Results such as the above enable us to approximate fairly rapidly to the constants 
of a frequency curve. 
As a special example, I take the following. In 1887, Herr H: pr Vries transferred 
several plants of Ranunculus bulbosus to his flower garden, and counted the petals 
of 222 of their flowers in the following year. He found (‘ Berichte der deutschen 
botanischen Gesellschaft,’ Jahrg. 12, pp. 203-4, 1894) : 
Petals4 ou ga te: 5 6 eu 8 g 10 
Frequency. . . 133 55 23 7 2 yy 
Now the series here proceeds by discrete units, and corresponds probably to a hyper- 
geometrical series, but remembering how closely the results of tossing ten coins can 
be represented by a normal frequency curve, I was not without hope that the areas of 
a skew frequency curve would give results close to these numbers. The buttercups 
start with 5 petals and run to 10, I therefore took my origin at 4°5 and determined 
the constants to a second approximation in the manner above indicated. There 
resulted, 
Y = 21122547 (73253 — oP, 
a curve of Type I., with limited range, the asymptotic ordinate being at 4°5 petals, 
or practically a distribution ranging from 5 to 11 petals. 
Calculating the areas, there results, 
Petals a. ores 5 6 7 9 10 11 
F J Theory. 5 6 | ROS 48°5 DRG 9°6 374 8 2, 
requency ; i 
LObservation . 133 55 23 7 2 2 0 
The agreement here is very satisfactory considering the comparative paucity of the 
observations.t The results are exhibited by curve and histogram, Plate 15, fig. 14; the 
two points on the “observation curve” corresponding to five and six petals are 
deduced from the areas given by the statistics by the same percentage reduction as 
* Another very serviceable formula is due to Scuuémitcu. It gives the area of the “tail’’ of 
y = ye Pe- 7 from g =z to «= ina rapidly converging series, #.c., 
area = Joe Pe” { je ees te = ON L (i+) ax + ke. \ : 
1 qe +l (a+ 1 (ye+2) (ye +1) (qe + 2) (ye +8) 
9 


+ 2048 tosses of 10 shillings at a time gave a mean 3 per cent. deviation between theory and 
experiment, 100 tosses gaye avout 9 per cent. The above series corresponds to about 7-2 per cent., and 
thus is quite within the range of accuracy of coin-tossing experiments, 
MDCCCXCV.— A. 3 F 
