40 
On the Variations in Personal Equation 
where negative values of the subscript of p and are to be treated as positive : 
e.g. iik = \,n = 5, j - 1, then p^+j-n = P-s = pi- 
We are again supposing that this secular change can be represented by y =f{t), 
a polynomial of degree ?(. in t, but we cannot expect that after removing a parabola 
of even 5th or 6th order*, the residuals Fj, Fg, ... F<,... will be mutually at 
random in time or space; if we anticipate correlation between F< and Yt+k< 
we must also be prepared for correlation between Yf and Yi+k^-^, and in any case 
the correlation between Yt and F, or p'k+j-n where j = n — k, will be unity. 
Hence we cannot make the assumptions of the first problem (that xyPp = 0, etc.), 
in fact 
^'A„r,A„r,,,is not equal to ry^y^^^. 
Now consider the use which may be made of equations (xviii) and (xix). If 
the values of the p^'s have been calculated from the crude values of the variate, 
the (juickest method of finding the correlations of differences nRk is not by direct 
calculation but by putting these known values of the ppS into the right hand side 
of (xviii). Then using (xix) we have a number of equations connecting the 
Pj, ""'s, and the question that at once arises is whether there are sufficient 
equations to determine these coefficients ? It will be seen at once that there 
cannot be ; if we are proceeding to ?;th differences, we can obtain q equations by 
putting k = l, 2,...q, but these will contain coefficients pi'"', to pn+,/'^^ ', in fact 
n more equations are required. By using the appropriate equations for the 
Product Moments and for the Standai'd Deviation of Hth differences corresponding 
to (xiv), (xv) and (xvi) we could obtain one further equation, but at the same 
time we introduce one further unknown, the standard deviation of the residuals. 
That these equations will be indeterminate, can be seen from another stand- 
point: the ?;th difference correlation equations (xviii) and (xix) will be satisfied 
not only by the pp's and pp^'^^'s as defined above, but by the correlation of the 
residuals left after the ordinates of a parabola of any order less than n, have been 
subtracted from the crude observations. Nor can further equations for the 
/3p<"''s, be obtained by proceeding to n + l, or higher differences; the further 
relations obtained will not be independent, for example 
T) 1 + 2 n-Ri ?i-^2 
2(1-, A) ' 
The possible application of these difference correlation equations is considered 
in the next .section. 
(b) The Application of the Results of tlie preceding Section. 
Although the correlation of differences does not appear to provide a general 
method for obtaining the correlation of successive values of a variate after secular 
changes have been removed, the equations (xviii) and (xix) will be found of con- 
siderable assistance in certain cases. 
* Tlie figures will probably not warrant the taking of differences of much higher orders than 
5th or 0th. 
