462 
PROFESSOR K. PEAR30X AND MISS A. LEE OX THE DISTRIBUTION 
As a rule there will exist for most triplets of stations a certain height, which mav 
be termed the balance height, bj which we are to understand that if the barometer 
stand at the balance height at two of the stations, its most probable value at the third 
correlated station is the balance height also. The balance height is at once found by 
putting A = — m^, h^ = — m.^ in the formula for regression and 
finding If we put Hgo = Hl = Hs., in the formula 021 p. 460, we are led to H 5 = oc 
for the balance height ; this is probably veiy far from being the true balance height, 
and for the simple reason that the factors, Oh of Hso and Hl on p. 460, are only 
approximate. 
If we found a Yorkshire station which had the correlation 0 ' 93“3 with both Laudale 
and Southampton, its balance height would be given by the formula, on p. 459 ; and 
supposing it to he on, or nearly on, the same generalised isobar as Stonyhurst, i.e., 
to have nearly the same mean and standard-deviation, then its balance heioht with 
Laudale and Southampton would be 29''-86S. An eimor of between O'OOl and 0-002 
in one of the factors, 0'5 of the Stonyhurst-Laudale-Southampton linear relation¬ 
ship, on p. 460, would thus have reduced the balance height of those stations from 
CO to about 29"-9. 
The general expi-essiou lor the balance height of a station 1 , with regard to 
stations 2 and 3, is given by 
+ (La - LsLd ^ - (1 - 
Hj (baiaiice) =- 
('22 ~ “ + (La “ - (1 *“ ''^a") — 
cr.2 a o cr. 
Hence if the three stations lie on the same generalised isobar, since rn^, m.i, rn.. 
and 0-3 have very approximately the same value, the balance height will be 
the mean height, and the same for every station with regard to the other paii’. 
One further general proposition may be noted before we leave the special case of 
prediction from two correlated stations. Suppose the barometer constant at one station, 
then the coefficient of correlation between the heights at the other two is given by"^ 
L3 = 
1 ’,.' 
'' 2.3 Ls'L 
v/1 - L?v/1-bjd’ 
_ ^'13 
x/l ^1- ’ 
_L2yy_Ly2s___ 
1 - ^ 13 “ - ^ 23 - ’ 
fo]- a selected height at the first station, 
for a selected height at the second station, 
for a selected height at the tliird station. 
* Ibese values bave been termed by Mr. G. U. Yule nett coefficients of correlation, to distinguish 
them from r^g, and r^o, which he terms gross coefficients. The difference would, perhaps, be 
best expressed mathematically by the use of such terms as joartial correlation coefficient and total 
correlation coefficient, the foi’mer being the value of the coefficient when one variable is not allowed to 
vary, and the latter when it is. They are at once obtainable from the general expression on p. 287 of 
the Memoir, ‘ Phil. Trans.,’ A, vol. 187, by putting, say x = const, and remembering that the coefficient 
of coia-elation for y and 2 iu P = P,|e-i(<- 2 ;'-- +"s--) is 
