OF FREQUENCY OF THE BAROMETRIC HEIGHT AT DIVERS STATIONS. 465 
13, Case (iii .).—Prediction from three Correlated Stations. 
The general formulas are given on p. 294 of the memoir previously cited. We do 
not reproduce them here, as v/e have no numerical data available at present by 
which to illustrate them. We will, however, consider a special application similar 
to that dealt with under Case (ii.). 
Suppose it possible to select three stations so situated round a fourth that the 
three stations have equal correlation r with each other, and each a correlation p 
with the fourth. The latter being marked with the subscript 1 in the formulae, the 
following are the proper relations which may be obtained after some algebraical 
reductions from the general case above referred to. 
X 
1 
pCTl j'h I ^3 I 
1 + 2r \ o-j cTg 
with a probable deviation 0*6745 /\^] 
3p^ 
1 + 2r’ 
po'2 -^6 , ('■ — P'~) ^2 / /ts I f ' 
+ 
I + 2r cTj ' 1 + r — 2p“ \ cr. 
+ 
a 
4 / 
with a probable deviation 0*6745 A/— - 
^ y 1 + r- 2p2 
Here the closest prediction will be obtained, if we select stations, if possible, such 
tliat p= (^i _|_ 27 ’), which is the point at which correlation passes into causation. 
In this case we find 
with vanishing probable deviations. In the first case, if p^>r, which can easily be 
true for correlated stations within 300 miles of each other ; in the second case 
always, a rise of the barometer at two of the “ outer ” stations for a steady barometer 
at the “ inner ” or first station marks a fall at the fourth station and vice versa. 
Now, it is not contended that four stations can be found for which exactly = 7’^3 
= and r^^ = r^^ — r^^, still less that it is possible to make 1 -f 2r = 3p^. But it 
is suggested that with the values of the barometeric correlation coefficients such as 
we have found in the British Isles approximations to these relations can be found for 
selected stations, and that such stations are what we require for close prediction or 
interpolation. Further, such principles as we have noted with regard to the relation 
Laudale is a low barometer at Soathampton, and vice versa. Or, the rule (by differentiation) may be 
stated for steady barometer at Stonyhurst a rise at Laudale indicates a fall at Southampton, and vice 
versa. 
3 O 
VOL. CXC.'—A. 
