K. Pkarson 17 



A litlK' coiisidoratidii will fri'(|iii'iitly Icad Ui llie selection of a curve witli as few 

 or oveii fewer coiistants giviiig a f';ir bcttcr fit. 



Thiole's observatioiis are giveu in § 6. Wc shall attenipt to fit a parabola- 

 series to Thiuic's trapezoidal polygon bctween a'=(J and « = 20, i.e. we shall 

 take 2^=14. 



The origiu tor moiiunit.s was takun at .«=13, and the siiceessivc niorrients 

 in„ixi, »ififJ'2, vi„fi.,', )ii„Hi, »'o(".i' '»'id )ii„/J-,! caiculati'd tor the System of trapezia fniia 

 the couceutiated ^'.s hy the lorniuhe 



/ii' = l'i', /i/ = ('/ + j'./c' + ^^c\ 



fJ-2 = '■■/ + ic", /X5' = v^' + ^ v^c- + i j// c\ 



fJ-s =W + ^ l'i' C\ Me = ''/ + 5 W C- + l'o C' + ^V C'^, 



where c is the base element, or in our case unity, and 



These formula; are deduccd in my memoir on " Ökew Variation in Homo- 

 geneous Material," Fhü. Trans. Vol. ISG, A, p. 349*. See also § 10 above. 



Theu\s = ^ was caleulated, and the following value.s obtained : 



X, = - •162,857, X, = -114,748, 

 X3 = - -033,778, X4 = -030,712, 

 X^ = - -010,204, X„ = -012,141. 



In addition we have y„ = 3.5-7l43, 



Mean a,'= 11-86. 



From these values the es were caleulated and the following series of parabolas 

 obtained, x being measiired from the mid-range : 



(i) y = 3.5-7143 {1 - '488,571 (x/l)}, 



(ii) y = 35-7143 [1-81!),(;94 - -488,571 (*•//) - 2-45;),082 {.i-;i.)-], 



(iii) y = 35-7143 {1-819,694 - 2-166,885 (a-ß) - 2-459,082 (.■r/0=+ 2-797,191 (a;/Z>'), 



(iv) y = 35-7143 {2-086,513 - 2-166,885 (x/l) - 5-127,275 (^vßf + 2-797,191 {.v/iy 



+ 3-112,892 (*-/01, ■ 

 (v) ?/ = 35-7143 {2-086.513 - 4-026,295 (a-/0- 5-127,275 (.r//f+ 11 -474,432 («/O'' 



+ 3-112,892 {xjiy - 7-809,518 {xßf}, 

 (vi) (/ = 35-7143 12041,057 -4-026,295 (.-•//) -4-172,701 (,-/0-+ 11-474,432 («/O» 



+ -249,l70(,,7'/y - 7-809,518 {.vßf + 2-100,062 (xß^'}. 



* These are not the proper formultc if we coiisidered Thiele's observations as the areas of a frequency 

 curve, but what we are here doing is to fit a series of curves as closely as possible to a polygonal area. 



Biometrika 11 i 



