512 
were unequally affected by the great weather variations 
of pressure. They showed a quite irregular variation 
from hour to hour. Thus his laborious effort, in many 
ways well planned, was fruitless. He concluded that his 
data were insufficient to determine L2, and hence that 
the attempts by Laplace and Bouvard were still less 
adequate. He urged that hourly readings of the barom- 
eter should be taken, so that in time, from a long series, 
L, might be determined.? 
The Tropical Lunar Air Tide. When Hisenlohr wrote 
(1843), LZ. had already been determined from tropical 
pressure data, though the result was not published 
until 1847 [91]. Around 1840, several British colonial 
observatories, magnetic and meteorological, were set up 
under Sabine’s leadership. In 1842 the director (Lefroy) 
of the St. Helena observatory successfully used his 
seventeen months’ (August 1840 to December 1841) 
bihourly week-day barometric readings to determine Lp. 
His successor Smythe, with Sabine, confirmed the deter- 
mination from three more years’ data, October 1842 to 
September 1845. Sabine even attempted to find how the 
air tide varies with the lunar distance [91]. 
In 1852 the director (Elliot) of the Singapore obser- 
vatory determined L, there [61] from five years’ data, 
1841 to 1845. 
Later, when the Batavia observatory was established 
in 1866, these results stimulated its directors [22, 23, 
96-99] to determine LZ, from their hourly barometric 
data, recorded photographically. Bergsma published 
the first results, for 1866 to 1868, in 1871. By 1905, Le 
at Batavia was really well determined from 350,000 ob- 
servations covering forty years. 
The Lunar Air Tide Outside the Tropics. Laplace 
(74a, c] stressed the need, in deriving results from ob- 
servations, to determine the probability that the error 
lies within narrow known limits; without so doing, he 
said, one risks presenting the effects of irregular causes 
as laws of nature, ‘‘as has often happened in meteorol- 
ogy.” This need has been overlooked or neglected by a 
multitude of those who, before and since his time, have 
vainly sought for lunar monthly meteorological varia- 
tions. Eisenlohr, from 1833, was among these; but only 
a few of them have, like him, engaged in the more 
hopeful but still perilous search for lunar daily meteor- 
ological variations, in particular for Lo. Of these few, 
some, like Kreil in 1841, or later Bouquet de la Grye, 
used quite inadequate data’—for one year only, or even 
for five years, like Neumayer [84], who in 1867 failed 
to obtain consistent results from five years’ hourly data 
(1858 to 1863) for Melbourne (lat 38°S). Even Airy [1], 
who used 180,000 hourly values for Greenwich (for 
twenty years, 1854 to 1873) unwisely concluded in 
1877 that “we can assert positively that there is no 
trace of lunar tide in the atmosphere.” Bornstein [24]! 
in 1891 used only four years’ data for Berlin and Vienna 
2. Not until 1945 was a further and successful attempt made 
to determine Lz at Paris. The results are not yet published. 
3. The method of computation is also important, as well as 
the amount of data, for the success of a determination. 
4. See also [12, pp. 39, 401. 
DYNAMICS OF THE ATMOSPHERE 
and Hamburg; for Keitum (lat 55°) he used ten years’ 
hourly data (1878 to 1887), but found ‘“‘no trace of a 
semidiurnal variation such as a lunar tide would pro- 
duce”; he thought, however, he had found a definite 
lunar diurnal variation (not, like L;, reversed fort- 
nightly; see p. 511). Bartels [12] m 1927 showed that 
these conclusions completely misinterpreted the actual 
results that Bérnstein had obtained, and that his sup- 
posed lunar diurnal variation was a purely chance ef- 
feet, whereas LZ. was contained in his curves, though 
it was ill-determined. The misinterpretation sprang 
directly from the neglect of Laplace’s advice to con- 
sider the probable accuracy of the results. 
Morano [83] in 1899 used four years’ data (1891 to 
1894) for Rome (lat 42°N), although Neumayer [84] had 
failed, by the same method applied to five years’ data 
for Melbourne, in a rather lower latitude, to obtain a 
reliable result. The validity of Morano’s result, which is 
probably near the true value of LZ. at Rome, remained 
uncertain. 
In 1918 a new attempt [28] succeeded in determining 
L, from the Greenwich hourly data, by then available 
for sixty-four years. Two-thirds of the material was re- 
jected; the only days used were those on which the 
barometric range did not exceed 0.1 in. This was the 
first certainly valid nontropical determination of Lp. 
This investigation was planned and undertaken with 
guidance [27], in a very simple way, from the theory of 
random errors. According to this theory, if a simgle ob- 
servation is subject to an accidental error e, the prob- 
able random error of the sum of N such (independent) 
observations is e+/ N, and that of their mean is e/+/N. 
The moon produces a systematic (though very small) 
semidiurnal variation of the barometer; if this is com- 
bined from N lunar days’ observations (each in the 
form of a sequence of lunar hourly values), by forming 
the sum of the values for each lunar hour, the sequence 
of sums will contain N times the lunar daily variation. 
It will, however, be affected also by the other causes of 
variation, particularly, outside the tropics, by the suc- 
cession of cyclones and anticyclones. If these produce 
an average random departure e of any hourly value 
from the long-term barometric mean, they will con- 
tribute to each lunar hourly sum of N hourly values a 
random contribution of the order e./N. As N is in- 
creased, the regular lunar daily variation in the lunar 
hourly sequence of sums will increase proportionately 
to N, and the random contributions will increase, but 
proportionately only to ~/N-. Thus, although e greatly 
exceeds the range of the lunar air tide at Greenwich, 
the systematic tidal effect will altogether overpower 
the random contribution if N is taken large enough. In 
the sequence of lunar hourly means, [2 is independent 
of N, whereas the random errors are of the order 
e/VN. 
As the Greenwich data were used only for days of 
barometric range 0.1 in. or less, the average random 
departures e from each day’s mean might be estimated 
as 0.01 in. Since N was 6457, e/~/N would be about 
0.00012 in. 
