86 
Proceedings of the Royal Society of Edinburgh. [Sess. 
at the 9-k. level and the corresponding changes at the surface. The 
changes which he dealt with were chronological, and I have extended the 
conclusion in applying it to topographical differences. This extension is 
justified if the places between which the differences are to be taken are 
sufficiently close together to be influenced by the same barometric system, 
and if the chronological sequence is followed in individual cases. That the 
latter condition is generally satisfied is shown by the high correlation 
coefficient between the variations at 9 k. and at the surface. 
The conclusion as to the relation between temperature and pressure in 
the upper air which is drawn from this table is supported by the gradual 
evolution of meteorological ideas on the subject. Originally it was assumed 
that high pressure meant relatively dense air and low pressure relatively 
light air from the surface upwards. Sometimes temperature and sometimes 
moisture was held accountable for the levity; but the view first put 
forward by von Hann that, in ordinary circumstances, the air over high 
pressure is warmer than that over low pressure has gradually developed 
until it may now be regarded as an accepted principle in meteorology. It 
is borne out by the simultaneous soundings which have occasionally been 
obtained from places within the same barometric system ; and apparently 
the disturbances in the specified order are mostly confined to the lowest 
reaches of the atmosphere. This last point also is well illustrated by the 
figures of the table, which show a gradual falling off, on the average, of the 
temperature differences in the lowest three kilometres. 
Lemma 2. 
The average horizontal circulation in the Northern hemisphere in 
January between 4 kilometres and 8 kilometres consists of a figure-of- 
eight orbit from west to east along isobars round the pole, with lobes 
over the continents and bights over the oceans. 
The average circulation at the surface is the resultant of the circula- 
tion at 4 kilometres combined with a circulation in the opposite direction 
of similar shape due to the distribution of temperature near the surface. 
[L. Teisserenc de Bort, Ann. du Bureau Central Meteorologique, 1887 ; 
and W. N. Shaw, Proc. Boy. Soc., vol. lxxiv. p. 20, 1904.] 
This lemma is introduced in order to supply the reader with a suitable 
general picture of the atmospheric circulation in the upper air, and the 
modification which it must undergo in the lowest layers in consequence of 
the distribution of temperature near the surface. As will be seen from 
Proposition 2, which follows, the similarity of pressure-distribution at all 
heights depends upon the equality of A 6/0 and Ap/p. Consequently, a 
