60 
On the Vanations m Personal Equation 
coefficient for the combined series corresponding to of the individual series, is 
given by 
Ri"= = — ,-- — - • — — (xxvi) bis, 
V \>H ) [m ) 
and taking p/, o-/ and aJ calculated from Equations (xxiii), (xxiv) and (xxv) we 
find that 
R/' = + -48922 + -01623. 
Then R/', R/'... Rj.," can be found by the method of Problem 2, p. 41; or 
again using the value of Ro" found from the first difference equations, we may 
proceed to second differences as in Problem 3, and so obtain R:,"...Rj2". In this 
particular case there is no need to use the second difference equations* but the 
values of the Rit"'s have been worked out by both methods, as numerical examples 
of the theory of Sections V (a) and (b). A comparison of the values given in 
Table VI shows that there is no significant difference between the results of the 
two methods -f-, and the agreement found earlier in this section between the 
values of R^/ calculated directly and those found from the difference equations, 
warrants confidence in the results for R^/'. Although the negative values of R^" 
are probably too large for the higher values of k (just as the later positive values 
of "Rk in Table V, row 7, were too small), there is no doubt, I think, that the 
correlation of the successive observations freed from the linear sessional change, 
does become negative at k = 5, 6 or 7 and remain negative for the higher values 
of k. A word of qualification is necessary ; the linear sessional change to be 
removed has been represented by the line " best " fitting the Jir'st 50 observations 
of each series, and a glance at Figure 4 shows that the mean values of the later 
observations in the series of 63 would lie well oft' this line because of the parabolic 
form of the sessional change ; the negative values found for Rg", R/', etc. may 
probably be largely accounted for by this fact. A more satisfactoi-y approximation 
to the correlation of successive judgments freed from sessional change will be 
obtained in Section XI below. 
As o-j = 0-] ^2 (1 — pi), referred to on p. 55 above, gives the standard deviation 
of the first differences of consecutive judgments in a single series, we shall have as 
a corresponding measure for the combined twenty series 
Ss = S,' V2(l-R/). 
For the Trisection Experiment 
= -0732. 
• 
* To get an idea of the order of the terms Qi. and //- which are being neglected, the values were 
calculated for two values of k, with the result 
- i'--!, Q^ + 62= --0000010G4) 
7.=9, ft. + P= + -000000192) ' 
1 - 
t The probable errors in the Table have been calculated from the usual formula e= ±-6745 — , 
and do not cover the errors arising from the method of approximation. 
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