118 Dr. C. Chree on the Law 
and Q given by 
8H/H=(3/2A){ (6174/7) ry4 (7? — 7,7) + (672/172)rg' (7 — 1717) | 
+ (82,/r,)7: rr —12 i 
dP = (3/A){6r,/7,) 7 4(%2' —7,') + (672/72) 72'(7'3° 11°) | 
+ (67,/7, 7s" (01,° To) 
SQ = —3(072127r,7/A){ (80/7) 7? (12? — 757) + (842/19)? (7,2 — 7,2) 7 
+ (6r,/7,)7, (7 — 72 
where | 
A=(r2—r2)(r2—r2)\(r2—"Y). (9) 
We shall suppose 7, the smallest, and 7; the largest of the 
three distances. 
If 57, 572, 673 are the corrections required to the values 
accepted for 7, 72, and 73, then 6H, 6P, and 6Q are the con- 
sequent corrections necessary to the values calculated for 
H, P, and Q. Considering, for example, the case where a 
correction is necessary to only one of the distances, say 7, 
we find 
6H/H=46P/(r.? + 7,7) = —436Q/r,77,? 
= —3(6r,/7,)ri={ (7? —19") (r?—7,”)}. (20) 
- The corrections to H, P, and Q thus depend not merely on 
the size of 6r, and on 7, but also—and that most materially— 
on the size of 72 and 73. In particular, if either or both of 
the quantities 7;~7, and r,~7; is small, a slight error in 7, 
may be decidedly serious. 
§ 5. To see the full significance of this result, let us con- 
sider the formule which take the place of (16) when 
(i.) P and Q are both negligible and observations are taken 
at the single distance 7, ; 
(ii.) Q is negligible, and P is eliminated by observations at 
the two distances 7, and 1. 
~ The formulee in question are 
for fi.) SHH =—(8/2)ér/r, 2... ee 
2 a 2 or 12 
(ii) pH/H=s { 2 
Sey ee 
Y, Ue) lip ie) ie) r, 
Comparing (21) and (22) with (16), we see that the errors 
in H answering to the same error in 7; stand to one another 
in the following ratio :— 
P and Q 
negligible. Q negligible. Neither negligible. 
1 2 rel (n2—re?) 2 {r2/(rP—re? {2 | (92 — 192) }- 
