12 On Ihe Systimatic Fitting of Cnrves 



Of courso if the \ determinant and its minore were once worked out, say for 

 the first six parabohw and tor tlie ciistoinary ränge of values of m, we should have 

 no niore labour, but meanwhile it seenis to nie that the method of least Squares 

 miist for practical piirposes be laid on one side even for jiarabolic curvcs oxcept 

 in the jiiinple cases of those of the tiret, second anil tiiird orders. But if the 

 method of least Squares be of smail practical use even in this one of the simplest 

 cascs of curvo-fitting, it may be (piestioned whether it is not better to adopt the 

 uniform process of monients thruugliont. 



(12) Lct US now apply the method of monients to the parabolic curve 



y = e„ + e, a; + ßj ar* + . . . + e„_,a:"~' 



for which the expansion by Maclaurin's Theorem is exact. Let 21 be the ränge 

 for which this curve is to be fitted to the observations, and let us take the origin 

 at the mid-point of the ränge. 



Let m„ be the area and ?«,, nu ... 7»„_i the first n moments of the Observation 

 poIygon about the axis of y, i.e. the perpendicular to the ränge at its mid-point. 

 Let US write nto = 2lxy^,so that y„ is the mean value of the ordinate. Then the 

 curve to be fitted mav bc writtcn in the form 



y = y«(eo + e,'''^+e.('jj +... + e„_, [j] j 

 iMiilt iply by f -T J and iutegrate from x = l to .t= - l : 



l-(-l)" 

 + ... + — ^ — ^ 



2r+S 2 2r + n 



If we multiply by («//)"■+' and iutegrate, 



l+(-l)" 



--/^-''^' = 2i/»^{27T-3 + 24-5^--^ 



2r + n + 1 



It is obvious that the even e's will be givon in terms of the even moments and 

 the odd e's in terms of the odd monients by tucj independent series of cquations. 



Let US write \, = mg:(7)ij'), thus X« = 1. 



Then X,= 60 + ^6^ + ^,6, + ... 



\ = ieo + iei + ie,+ ... 



K = },eo + \ei + ijet + ... 



^^5 = 761 + ie3 + -^et+ ... 



