MATHEMATICS: J. R. MINER 
9! 
A NOTE ON THE FITTING OF PARABOLAS 
By John Rice Miner 
BIOLOGICAL LABORATORY. MAINE AGRICULTURAL EXPERIMENT STATIONi 
Communicated by Raymond Pearl, November 28, 1916 
The formulae given by Pearson^ (pp. 12-16) and Elderton^ (pp. 30-31) 
for the fitting of parabolas by the method of moments assume the origin 
at the mid-point of the range. It being often more convenient to take 
the origin one unit below the first ordinate, as in working by the method 
of least squares, I have, at the suggestion of Dr. Raymond Pearl, 
worked out the formulae which result from such choice of origin. 
Let / be the range for which the parabola 
y = Co-\- CiX + C2X^ +....+ CnX"" 
is to be fitted to the observations, and = S{yxr) where the summa- 
tion includes the values of x and y for every observation. 
Then 
Mr= (co + CiX + C2X^ + . . . . CnX"") X^'dx 
r+l r+2 
+ [(/ + *)'+"+'- (§)'+"+'] 
r -\- n +1 
Substituting r = 0, 1, 2, . . . . n in this formula we have n +1 
simultaneous equations from the solution of which we may express the 
c's in terms of the moments and certain functions of /. 
(i) . y = co + cix, 
Co = KiMo — K2.M1, 
Ci = - K2M0 + iTgMi, 
where 
K, = 1 (4/2 + 6/ + 3), K2^j^{l+ 1), Ks = 12/P. 
(ii) . y = Co + cix + C2xK 
Co = KMo - K,Mi + KeM2, 
ci = - KMo + KMx ~ KMi, 
C2 = KeMo - KsMi + K,M2, 
