10 



Oll Ihr Systemalic Fittiiig of Ciirves 



a iiiiiiiiiiuin for values of c», Ci, C5...c„_i we casily find the type ciiuations 



f,' = c„X,' + c,V + CqV + . . . + c„_,\'„ 

 i// = CoX./ + CiX/ + CjX/ + ... + c„_,X'„+, 



l» n_l — CjX „_, + Cj X „ + CjX ,n-i + ... +Cn_iX5„_i. 



These are n equations to find the n constants. 



Now in any curvc-fitting with 11 constants to bc found wc must detcrminc the 

 area and nionients v„', Vi, v^ ... "',1-1 ! but we find that the inethod of Icast Squares 

 iuvolves also the discovery of the 2h qnantitics Xo', X,', Xj' ... X'j„_,. 



If as is very usually the case the observations are takcn at cqual intervals and 

 there are m of them, wc can by choosing this interval as our unit and a proper 

 origin, write 



X/ = :£ (./■■•) = r + 2-- + s-- + . . . + Mi--, 



or X/ is the siini vi' the rth powers of the first m natural numbers. Thus the 

 values of X/ for .suecessive values of »rt can be tabled once and for all. This is 

 done for 7/i.= l to 20 and ?• = 1 tu 7 in the accompanyiiig 'l'able. 



Table of the Sinns of the first seven Powers of the Natural Numbers*. 



* Mr W. Palin Klderton lias calcnlated an ext«nded table of this kind, giving the sums of the 7th 

 powur» of all the natural numbers up to 100. I hopo it miiy bc possible eventually to rcproducc and 

 distribute this much eulargcd table. 



