38 On the Variations in Personal Equation 
V 
(c) % (xt+h'''t+h^k) = vx.-cpjccrj' for all values of /i small compared with v, 
tSimilar relations will hold for the residuals A'^ and Y. 
Then a little consideration shews that the sum df the coefficients of the 
products of the x's and t/'s whose indices differ by p in the expression 
XiXf ^iiVt or (1 - e)" Xn+t (1 - e)" \j,+t 
is the coefficient of v x x,,p,^,a.xa-,, in the product moment 
S A,i,r, . A,i?y, ; call this coefficient (/^j. 
i = \ 
Now e'' operating on gives 
» Vn+t 1, yn+t-i- 
and if {n +t — r)~ {u^ t —r')=p, then r' ~r —p; hence is the sum of the 
coefficients of the products e{ e/ in the expansion of (1 — 6])" (1 — 6.,)" for which 
r — r = p, or the coefficient of e'' in 
(1-6)", . 
orof6«+Pin (_ i)" (i _ g)-^"^ 
so that o« = (- 1 rr, r. • 
Hence finally writing j = n+ we have 
1 " 2?!. ' 
where negative values of the subscript of p imply that the subscript of x is less 
than that of e.g. ^i/P-p is the correlation between Xt and yt+p- 
Similarly for the standard deviations of the Jith differences 
- 3, ('^..2")' = <,!„ (- 1)"' (2ir37yrj! (™). 
and for the correlation between the differences 
2n ' 
— , ^ ■- — ...(xvii). 
VL-r/ (2»-i)!j!-^Ml/ro^ (2«,-j)!i!''^'-«| 
The correlation of the ?ith forward differences of the residuals Xt and Yf or 
will equal an exactly similar expression to the last, in which yj-p, xP ^^'i yP a-re 
