92 On the Variations in Personal Equation 
we have as equations of condition for a least square solution 
8jj + rj'hq + kq„r„''-'^ Sr = R/ - p(, - (/o?;*, for ^- = 1, 2, . . . 13. 
Using the values of p^, q„ and given above, the corrections hp, 8q and 8r 
wei-e calculated and gave finally as the best fitting numerical equation, 
Rk = •1524 + -6817 (-710.-) (xlix). 
TABLE XVIII. 
Values of the for Trisection Experiments. 
1 
2 
3 
4 
5 
6 
7 
8 
k 
(direct 
calculation) 
(from equation 
(xlix)) 
Difference 
Col. 2— 
Col. 3 
Probable 
Krror of 
Values obtained from 
(lii) on assumption of 
constancy of G^. 
(from equation 
(Ivi)) 
0 
+ ^834 
+ ■804 
1 
+ •625 
•637 
- ^012 
±•013 
■0773 
+ ■550 
■571 
2 
•523 
•497 
+ ^026 
+ ^016 
■0776 
•431 
■406 
3 
•388 
•397 
- •009 
+ ^018 
■0778 
•268 
•288 
4 
•315 
•326 
-■Oil 
+ ■OIQ 
•0781 
•183 
•205 
5 
•281 
•276 
+ ■005 
± ■oao 
•0778 
■142 
•146 
6 
•232 
•240 
- ^008 
+ ^020 
•0782 
•084 
•103 
7 
■222 
•215 
+ ^007 
+ ■oao 
•0782 
•071 
•074 
8 
•191 
■197 
- ^006 
± -021 
■0783 
•035 
■052 
9 
•165 
•184 
- ^019 
+ -021 
•0787 
•006 
•037 
10 
•183 
■175 
+ •008 
+ ^021 
•0802 
■031 
•026 
11 
•168 
■168 
•000 
•f- 021 
•0823 
•017 
•019 
12 
•172 
•164 
+ ^008 
+ -021 
•0834 
•023 
•013 
13 
+ •160 
+ ■160 
•000 
+ -021 
•0840 
+ ■009 
+ •009 
14 
■0840 
In the second column of Table XVIII are given the values of R^' taken from 
Table V and in the fifth column their probable errors ; the values of R/ given by 
equation (xlix) are in the third column, and in the fourth are the differences 
col. 2 — col. 8. It will be seen that the fit is a good one, the difference being only 
greater than the probable eri'or in the ease of R/. The points (k, Rjt') and the 
curve of (xlix) are shown in Figure 8 (p. 64). 
The problem before us is therefore this ; can we explain the constant p in 
equation (xlviii) in terms of the sessional changes? We have seen that the mean 
sessional change for the 20 series can be represented by a parabola of the second 
order, but we must allow for a different change in each series. Let us suppose that 
y=Mt) . 
will represent the sessional change in the ^^th Series after the secular term 
represented by the series mean has been removed, so that instead of equation 
(xlv) of p. 89, we have 
yt'=fp(t) + a, + f3,=fAt)+Y, (1), 
where Y, = a, + jS,. 
