OF FREQUENCY OF THE BAROMETRIC HEIGHT AT DIVERS STATIONS. 463 
The values of these expressions are peculiarly interesting, for they lead us to the 
general theorem that whenever the total correlation between two stations is less 
than the product of the total correlations at the other two pairs of stations, then 
the partial correlation between the latter two stations will be negative, i.e., for 
a o-iven value of the barometer at the first station, a risino- barometer at the second 
o 
station will on the average mark a falling barometer at the third station, and a 
falling barometer at the second station a rising barometer at the third. 
Owing to the generally high values of barometric correlation it is comparatively 
easy for for example, to be greater than ';’ 23 . Thus in the case where two 
of the stations have equal correlation with a third wm have, if — p, 'jq., = r^g = r, 
^’i3 — ^’12 — 
v/1 - 
and Tgo will always be negative if r" be > p. For example, we have Laudale and 
Southampton with a correlation 07572, while Stonyhurst and both these stations must 
have a correlation of about 0‘94 to 0'95. It follows, therefore, that the partial corre¬ 
lation of Laudale and Southampton with regard to Stonyhurst is negative, or for a 
constant value of the barometer at Stonyhurst rising at Laudale would in general 
denote falling at Southampton, and vice versa. This is best illustrated by noting 
that aSg being the mean height at Southampton for heights /q and /q at Stonyhurst 
and Laudale (.%, /^l and Ag being measured from the respective mem heights of the 
stations). 
•r., 
1 ■ 9 ’ 
- /fs + 
^'12 ^23^13 
or, in the special case approximately^ 
p — 'r TO, , r (I — p) c. ,, 
1 ~ ' ' 1 1.3 ^ 'h • 
I — cr.. i — ? (Ty 
Hence, if rjgr^., be > r.^g or r- > p, an increase of Ag for a constant /q means a 
decrease of yq. 
In the particular instance of correlation passing into causation referred to by 
us on p. 459, i.e., when 1 + p = 27’'^ we have 
r.^g- I , 
^13 — ^12 — F 
CTo 
X 2 = 
/q + 2r"Vq. 
Thus the partial correlation between the three stations is “ perfect,” but between 
the second and third it is negative. The ratio of the fall at the second to the lase 
at the third, for stationary barometer at the first, is crja-^. 
This principle, which at first sight appears rather paradoxical, namely, that at 
three stations, A, B, C, a rise at A will, on the average, be accompanied by a rise 
at C, but that a rise at A for a constant barometric lieight at B may, on the average 
