92 
MATHEMATICS: J. R, MINER 
where 
K, = (^l^ + 4/^ + 7/2 + 5/ + I) , iTs = ^ (/ + 1) (4/2 + 1 0/ + 5 ) , 
Ke = j (l' + 31 + K,=^ (16/2 + 30/ + 15), 
i:8 = ^(/+l), K, = m/IK 
(iii). 3; = + ^^lic + C2X^ + CzX^, 
Co = KioMo - KnMi + KnM2 ~ KuMs, 
ci= - KuMo + KuMi - Ki,M2 + KisM,, 
C2 = K12M0 - KibMi + KnM2 - KiMz, 
C3 = - KuMq + KiqMj - KisMz + Ki^Mz, 
where 
i^io = J_ (64/6 + 480/5 + 1680/4 + 2840/^ + 2460/^ + 1050/ +175), 
4/^ 
= 1^ (/ + 1) (16/4 + 96/3 _|_ 183/2 + 140/ 4. 35), 
= ^ (16/4 _^ 104/3 4. 192/2 _^ 140/ + 35), 
/^ 
i^i3 =^(l+l) (2/2+ 10/ + 5), 
i^,, = lA (16/4 + 72/3 _^ 120/2 _^ 84/ + 21), 
ir.s=^(/ + l)(3/2 + 7/ + ^), 
iTie = ^ (4/2 + 10/ + 5), iTiv = ^ (36/2 + 70/ + 35), 
iri8 = ^(/ + l), iri9 = 2800//^ 
The values of the K's for values of / up to 30 are given in Table 11. 
The fitting of the following observations, given by Thiele4 (p. 12) and 
used by Pearson2 to illustrate his formulae for fitting parabolas, will serve 
as an example. Table I shows the calculations to obtain the moments 
and the resulting parabolas. It is obvious that these data are in no 
way suited to graduation by parabolas, being really a unimodal fre- 
quency distribution. They will, however, serve for illustration of 
method. 
The origin for moments is taken at X = 6 and the successive moments 
corrected by Sheppard's formula (X)^ (p. 276). 
