456 
PEOFESSOR K. PEARSOE AND MISS A. LEE ON THE DISTRIBUTION 
at Babbacombe and Hillington. The probable deviations in the two cases are 
0''‘0401 and 0"‘0663. Thus, the prediction would probably be correct to within 
2 V of an inch. 
We should not propose, however, to base the prediction of the barometric height 
at one station on its correlation with the height at a single other station. We should 
think it desirable to apply the principles of multiple correlation, and endeavour by a 
suitable selection of stations to decrease the probable deviation of the array at the 
given station which corresponds to observed heights at the selected stations. We 
shall cite here the general formulae for the prediction of the height of the barometer 
from a knowledge of its heights at correlated stations. 
Let x„i be the probable height of the barometer at the station above its mean 
value for that station. Let r^m' be the coefficient of correlation between the and 
m'* stations, where in finding the correlated heights may be taken at different 
times, or if it seems desirable on different days. Let be the observed height at 
the station and cr,^, the standard deviation, both measured from the mean of that 
station, 
11. Case (i .).—Prediction from one Correlated^ Station only. 
a-] = r ,3 —Ag, with a probable deviation of 0'6745cri\/l — 
^ 2 
It is. clear that unless r^^ be very nearly unity, i.e., the stations very close, the 
predicted height v/ill be subject to a large probable deviation. For very close 
stations, such as London, Cambridge, Dover-Dungeness, where a rough investigation 
leads me to the conclusion that the correlation is as high as 0'998, we have the 
probable deviation about 0‘04487 X cr^, or about 0’015". In such cases the above 
formula will give fairly closely the height to be predicted at the first station. It 
may be used for purposes of interpolation. 
The approximate linearity of the “ regression ” leads us to an interesting projoerty 
of barometric correlation. There is a certain height of the barometer which, if it 
occurs at one station, will itself be the most probable height at the correlated station. 
This may be termed the balance height. This height may be easily found from the 
equation : 
m-, X. ■= 4- h-,, 
whence, if 
Pn = »’i20-i/o-2. h = (»H — ■'%)/(! — Pa), 
and the balance height 
— ~ Pi^ni^j(\ — 
Above and below the balance height the relative heights ot tlie two stations are 
reversed. If above the balance height tlie first station has a probable height of its 
barometer invariably higher than the observed height at the second station, then 
below the balance- height the probable height at the first station will be invariably 
lower than the observed height at the second station, and vice versd. 
