86 PROF. K. PEARSON ON THE MATHEMATICAL THEORT OF EVOLUTION. 
This curve is plotted out as the dark continuous line in Plate 1 , fig. 1 , and is 
clearly asymmetrical. 1 proceeded to calculate its first five moments in the analytical 
method suggested on p, 78 {a), each calculation being made twice independently. 
I took /i = 1 , and clearly a — 1000 . The moments were taken about the vertical 
through the point 0 , and were calculated by the aid of Table I. of the powers of the 
first 30 natural numbers given at the end of this memoir. The following results 
were obtained :— 
/x/ = 16-799 
= 304-923 
= 5,831-759 
— 116,061-435 
/X 5 ' = 2,385,609-719 
p-i', since h ■= 1, is clearly the distance of the centroid vertical of the frequency- 
curve from the origin 0, i.e. = g' of p. 77 (ii.). 
The moments about this centroid vertical were now calculated by aid of (1), p. 77. 
There resulted :— 
1 ^ 1 = 0 
= 22-716,599 
ix.^= — 53-874,770 
fx^= 1576-533,413 
lx-= — 9598-313,922 
= - 85-205,407 
Xg = — 7920-604,761 
where X^, Xg are given in terms of the /x’s by (22) of p. 84. 
Turning now to the fundamental nonic (29), let it be divided by 24, and written in 
the form 
Pi + + ^hP-^ + Vp + ajp/ + flgPa + «9 = 0. 
Then the coefficients a^, ct^, 
. were calculated, and the following values found :— 
a,. = 
«3 = 
= 
a- = 
«6 = 
«8 = 
Og - 
99-406 
4,353-742 
423,696 
3,702,933 
119,298,911 
1,232,409,400 
957,080,900 
24,451,990,000 
