16 On the Si/stemafic Fitfing qf Curves 



Hence we have 



eo = H (15X„-70X,+ 63XJ, e, = }^{- (35\,- 12GX, + DOX»), 



e, = J^(-5X,+ 42X,-45X4), e, = äj^^ {- 2l\+ OOX^-TTX,), 



e, = y{- (:}\, - 30^. + 35X,), es = -7j''- (15X,- TOX, + G3X,). 



Gase (vi). To fit a parabola of the sixth order to a series qf observations. 

 Let the curve be 



y = y» jeo + e, f + e, [ff + e, (?)' + e. (f )' + e, (f)' + e, (f )] . 



Then Xo= ^„ + ^^3+ ie^ + ^e,, X, = ^e, + ^63 + fßs, 



^! = Jeo + ^ej+ |e4 + ^e„ X, = ^e, + |e, + ^gj, 

 X-4 = ieo + jej+ ^e^+^V^e, X5 = |e, + ^«3 + 1^65, 

 ^ = f e„ + ^63 + jVe« + tV««- 

 Hence we find 

 eo = ih (3öX„-315X,+ 693X,- 429X,), e, = Vr (35X, -126X, + 99X,). 

 «! = 5i5(-35X„ + 567X, - USÖX, + lOOlX,), e, = ^«-(- 21X. + 9OX3 - TTX,), 

 «4 = ^s'W^ (7X„-135X.,+ 3S5X4- 273X,), 65 = W (l-'>^.- 70X3 + 63X5), 

 e6 = 4i"^(-5X«+10.5X,- 31ÖX,+ 231X6). 



(13) Illustration VI. In order to thoroughly tost the manner in which succes- 

 sive parabolas fit niore and niore closcly to a series of observations, I have taken as 

 a first illustration a very unproniir^ing series of observations given by Thiele in his 

 Foi'elaesninger over Almindeltg lagttagelsealaere (K^ifihcnhiivii, 1.S89), p. 12. I say 

 u)ipromising because the observations are not such sis one wouid in practice 

 attempt to fit with a parabolio cuive; thcy form a frequcncy distribntion for whieh 

 my skew frequency curve of limited ränge gives a very good fit as we have seen 

 above in § 6. But to fit thcse unproniising observations evcn approxiniately is of 

 great interest; the process shows us much inore clearly than would otherwise be 

 the case the struggles of the successive parabolas to get their points of inöexion 

 to the approxiniately corrcct pnsitions, and, to sjjcak anthroponiorphicall}', their 

 rather futile attenipts to bend theniselvcs into the shape of the Observation curve. 

 But a still morc important principle is illustratcd whcn we compare ihese 

 p.arabülas with the geueralised frequency curve, naniely that the nunibcr of 

 constants at our dispo.sal is no ineasure of the goodness of the fit. The skew 

 frequency curve with three coustants fits much better than the parabola of the 

 sixth Order with seven constants. Thus in fitting an empirical curve to 

 observations it is all-important to make a suitable choice of that curve; i.e. 

 to deterniine whether it shonid be algobraical, cxponential, trigononictrioal, etc. 

 There is indeed very little to justify the reailiness with which in practice a 

 parabola of one or another order is sclccted to describe the results of Observation. 



