I'karson 



11 



ll will l)c soeii at onec lluit \~' is very largo l'or any uuiiibcr of obsurvalioiis 

 greatiT tliaii 10. Even if \ve go as f'ar as X/, \ve shall, liowevcr, be ouly fitting 

 a paiabola <>(' the Üiird order, and tho four type oquations to be solved will b(; 

 found as a rule evcn in tliis case to be ratlier unmauageable. For parabolas of 

 the fourth, fifth and sixth Orders, the labour bccomes very severe. 



It is clcar liowcvur that if \ve fvaluatcd tlie (h/tiTniiaaiit 



X„ , X, , X._. , ... X „_j 



X, , Xo , Xj , .. . X „ 



\' < ^;i > '^11 ... X „-I-, 



X „ 



^ /i > 1^ n + 1 1 



X ..,), 



and its iniiiors for valucs of /* =1, 2, ... •"), (i, and of m froni 1 iip to 20, wo should 

 have a set of constants which wonld enablc u.s readily to find Co, Ci, ... as soon as 

 the Vo, Vi,...v'n^i were given. The arithmetical work to be once done would 

 be considerable, but it might be worth doing, .supposing the method of inoments 

 were not available with a simpler Solution. 



It niay bc notod that a considerable simplification of the least Square type 

 equation.s can be introduced if there be an odd numher of observations. Lct us 

 take the origin at the niiddle Observation, then cloarly 



X, = X3 = Xj = . . . = 0, 



or all the odd x sums vanish. Lot ns use undashed letters to denote moments 

 about the centre of the ränge, then we find our system of type equations breaks 

 up into two 



i'o = CuX,i + c.X,^ + CjXj + and v, = CiX. + C3X4 + CsX^ + 



Vi = CoXo + C2X4 + CjXs + i'3 = C1X4 + c^Xß + CsXg + 



Vi = C„\i + cAß + CjXg + V', = Ci\ + CjXa + CsXio + 



Our Table will now enable us to find X^ for an)- number of observations up to 

 41; and for parabolas of the third order only, we have simply to solve two sets of 

 linear equations, each of which contains only two unknowns. Thus the work 

 becomes extremely simple. This is in faet how the cubical paraboJa was fittcd by 

 the method of least Squares to the observations in our illustration in § 3 (Vol. i. 

 p. 280). 



On the other hand even with this choice of origin we require X„, X,o and X,, to 

 fit parabolas of the fourth, fifth and sixth orders ; and the sum of the lOth or 12th 

 powers of the natura! numbers* and simultaneous equations with three or four 

 variables and coefificients of very diverse niagnitudes are rather tronblcsome 

 matters to deal with. 



Ol course X,. may be caIculateJ from the Bernoulli mimber formula on ji. 286, Diumetiikn, Vol. i. 



2—2 



