192 
Sleiv Variation, a Rejoinder 
to place this new representation on the same footing as the symmetrical binomial 
to which Laplace approximated with the normal curve, I deduced as I had done for 
the symmetrical and the asymmetrical binomials, curves which gave the hyper- 
geometrical series and the sum of its terms as closely as Laplace's normal curve 
gave the symmetrical binomial. This is the complete history of the development 
of my skew curves. Before I proceed to discuss Ranke and Greiner's criticisms, 
I must remark that their attack on this point does not concern me only. Every 
practical statistician uses Laplace's representation of the point binomial by the 
1700 
1600 
1600 
1400 
1300 
1200. 
1100. 
1000 
■ 900 
ti 600 
700^ 
600 
500 
400 
300 
200 
100 
f 
4 
L 
\ 
L 
V- 
— / 
V 
, — 
H 
1 
— J 
, — 
—2 —10 1 2 3 4 6 6 
Number of Dice shmving 5 and 6 pips. 
Fig. 4. 5000 Throws with 12 Dice. 
O O Points of the Binomial + g)'- x 5000. 
Curve y = 1536-54 ( 1 + 1) ^ 
Origin at the mean 2. Mode of curve at .r= -i. 
Frequency of each number of 5 and 6 pips - 1 0 1 2 .3 
Ordinates of Binomial ... ... 0 .561 1346 1480 987 
Areas of Curve ... ... 557 1340 1493 982 
4 
444 
443 
5 
142 
139 
6 
33 
37 
