K. Pearson 
293 
constants will often give far less advantageous results than the choice of a more 
suitable form even with fewer constants. I will endeavour to illustrate this by the 
following system of frequencies due to data from a game of ' patience* ': 
Value of 
character 
5 
6 
7 
8 
9 
10 
11 
13 
13 
U 
15 
16 
17 
18 
19 
20 
21 
22-28 
Frequency 
3 
7 
35 
101 
89 
94 
70 
46 
30 
15 
4 
5 
1 
The possible range is 4 to 28. 
Now we find the mean of the character to be 11 '86, and let us assume the form 
of the curve to be 
2/ = yo(l+-) e «. 
Here the origin is at the mode or maximum ordinate which is at a distance ajp 
from the mean. We have thus otjly three constants p, a and at our disposal. 
We shall show in the sequel that we get a better fit than if we disposed of seven 
constants in a curve of the form 
2/ = tto + a-^X + tts*'^ + CliX^ + CliX* + Ctr^X^ + QeX^. . 
The data appear to give high contact at both ends, and therefore Sheppard's modi- 
fications would give the be.st values of the moments. But for the purposes of 
illusti'ation we will treat the data as giving a polygonal curve, and assume our 
object to be that of finding a curve going as close to this polygon as possible. 
Methods of finding the moments of a polygonal area will be given in the second 
part of this paper. Formulae for our present purpose will be found in Phil. Trans. 
Vol. 186, A., p. 350. There results for the moments about the mean in the present 
case 
/A2 = 4-3231, = 4-6804, = 59-683. 
Hence we deducef 
98-762 (l+^J 
The distance from the mean to the mode, which is the origin, is -5413. Thus 
the modal value is at 11-3187, and the range starts at 3-2931. 
The ordinates corresponding to the observations are given in column two of the 
Table in Art. 13 of this memoir. Fig. 3 shows a reasonable fit. The Table 
compares these results with the successive parabolas up to the sixth and shows 
how a well selected curve with three constants can easily be superior to one with 
seven constants J. 
* Thiele ; Forelaesninger over almindelig lafjttageheslaere, p. 12. 
+ The formulae for^j, a, and in terms of the moments are given, Phil. Trans., Vol. 18G, A, p. 373. 
J This point is of special importance, for objections have been raised against the skevc frequency 
curves just referred to on the ground that they give better fits than the normal curve because they have 
one or two more constants as the case may be. This is true, but they also give better fits than some 
other curves with double their number of constants ! 
