CORRECTION OF DATA FOR MEASUREMENT 25 



and the ratio of the 4tli moment to the square of the 2n(l is numerically 

 3 which indicates that/o(A'') is normal in form. 



A similar substitution of the 2nd approximation form for/; (A') in 

 Equation (11) yields a distribution fo{X) from whose moments we 

 deduce Equation (7). Use is made in this proof of the easily demon- 

 strated theorem that 



/ 



\[ i < j, where //r is thejth derivative of the normal law function. 



APPENDIX II 



It is well known that the normal law of distribution may result from 

 a system of n (w being large) causes each of which produces an incre- 

 ment AX measured from some fixed origin with a probability p = h and 

 no increment with a probability g = |. Furthermore the second ap- 

 proximation may result from a similar system in which p + q and n 

 is large. Under such systems of causes, the probabilities of the oc- 

 currences of n, 11 — 1, ... 3, 2, 1, increments are given by the suc- 

 cessive terms of the point binomial {p-\-qY- 



Let us assume that the symbols pr, qr, "r, AX and p/?, qE, n^, A.r 

 refer to the systems of causes controlling the product and errors 

 respectively. The probabilities of observed combinations wrAA^-f 

 WfiAx, (wr— l)AX + («£ — l)Ax, . . . are given by the successive terms 

 of the expansion {pT+qrY^ (pE + qEy^- Now for the special case 

 Pj~ = Pj, = p and AX'=Ax we have the resultant probability distribu- 

 tion {p^-qy^^"^ with skewness 



q-p 



X pq{nT-\-nE) 

 and standard deviation 



(To = \/pq{nT+nE). 



Now if p = q, the skewness ko is zero and the observed distribution 

 is more nearly normal than either component, and its standard devi- 

 ation 0- is the square root of the sum of the squares of <^t and ^e- 

 This result is similar to that given by Equation (4) of this paper. 



We may also consider by this method a case not treated in this 

 paper. When the skewness kr of the true values is equal to that 

 kE of the law of ertor, or, more particularly, when nT = nE = n, pr = 

 pE = p, qT = qE = q, P = q, we see that the observed distribution is given 



