18U Netv Tahles <>f thc /'robahl/K;/ htln/Kd 



This, ot' courso, is guesswoik. If thero are inore thaii three or four valiies of x 

 tu be dealt with, we can do our guessing by graphic methods. The values 

 of X should be plotted as oidinates, with a mark on each side to shew the probable 

 error: and we have then to draw a straight line which sliall make the error 

 as sniall as possiblo, allowance being made for tbe difterout valiies of the probable 

 error in the differeiit ordinates so plottuii. It niust be reinembered that the 

 crrors are not independent, but eorrelated : the correhition between the crrors in 

 any pair of values of a; being positive. 



If we want to procccd more rigidly, we miist use the actual values of x, 

 and take account of diff'eronces. This is practically equivalent to regardiug X as 

 an unknown function of some other quantity Y, whose values are normally distri- 

 btited ; the rclation between X and 1" being such that dX'dV is always positive. 



8. Ctilculation of ordinates. The orilinate Z of the curve of frequency is dXJdX. In most 

 cascs this can be calculated directly froin the data, by the first formula of (5) (takiiig « = J',) or 

 a siniilar formula. Whcre the difiFerences in u are too irregulär for this, bvit the diflFerences in .r 

 are comparatively regiilai-, we can u.se the data in the form shewn in § 5 : we have then 



Z= X.d^ (1 +a)ld.v. dsldX=Nz . dx/dX. 



Thc values of rf.i dX aro. given by the data, and the value.s of » are found froni Table II. 



(t. TestiiKj for normal didrihutiun. This is the purjjose for which tables 

 such as Table I. are most froijuently eniployed. It is not necessary to give 

 any examplcs here. 



10. Galculation oj correlation-volumen. The formula, given by Professor 

 Pearson*, for calculating the double-integral cxpressing normal correlation, in- 

 volves a factor 



_e-S(x- +!/'') 

 27r 



which is the product of the values of z as given by Table II. for .<• and for y 

 rcspectively, and therefore is easily fonnd from thnt Table. 



Construction of tlie Tables. 



11. The tabulated values are all given to seven dcciuial places (and, in the lattcr {wirt of 

 Tables I. and IL, to ten det-inial place.s). They were originally calculated to two or thrcc uiore 

 placas, the final figure.s being then corrected. Where the tinal figure Wfus doubtful, the value 

 was calculated specially. The dirt'erences are taken froni the largcr table, the last figure being 

 corrected : but doubtful values were not .sjiecially calculated. 



For constructing Table.s I. and II. up to :r=2'5(), the value.s of c were found for the inter- 

 uicdiate values -OGö, 'Olö, 'OSO... of .r by successive luultiplication ; each tentli value being 

 checked by Newniau's table t of e - *. The diflereuce.i being then ttvken, the vjxlues of i ( 1 + a) for 



• Phil. Tram, series A, Vol. I!t5, pp. 1 — il. 

 t Cumh. l'hil. Suc. Tram. Vol. xin. l't. :t. 



