514 
scale values for the pressure and speed variations asso- 
ciated with S» will be ten times greater than for Lo. 
A harmonic variation such as the nth term of (1) can 
be graphically represented by a curve, for example, the 
broken line for Lz in Fig. 2 or the curve for S» in Fig. 3a;° 
or it may be specified by a diagram which indicates the 
amplitude c, and phase yn, as m Fig. 3b [16]. This 
shows an origin O, a phase-reference line OX, and a 
line OC whose length, on the amplitude scale marked 
along OX, represents c, (in this case s2), and whose 
direction indicates, by the angle XOC, reckoned anti- 
clockwise, the phase y, (in this case cz). A circle may be 
drawn with O as centre, and graduated, as in Fig. 3b, 
to show the phase angle. 
A perpendicular axis OY may also be added, and the 
circle may also be graduated (clockwise from OY) in 
time measure ¢ = n6, at the rate 360° per interval 7'/n. 
For Fig. 3b, n = 2, so that 7'’/n is half a solar day or 
twelve solar hours, and the time measure is therefore 
30° per solar hour (or in a similar diagram for Lo, 30° 
per mean lunar hour, reckoned as 24 per lunar day). On 
this graduation, OC points to the time given by n6é = 
90° — yn, at which the harmonic variation has its first 
maximum in the interval 7. When n = 2, the diagram 
corresponds to an ordinary clock face, on which each 
hour corresponds to 30°; OC points to the times of 
maxima, morning and afternoon. When n = 1, the dia- 
gram may be likened to a 24-hour dial. When n = 3 or 
n = 4, the diagram will show only eight or six hours on 
its face or rim, at intervals of 45° or 60° respectively. 
Because of this alternative interpretation of the dia- 
gram, as a clock face indicating the amplitude and 
time(s) of maximum, the diagram is appropriately called 
a harmonic dial [30]. 
Any part of a harmonic dial not needed in a particular 
case may be omitted; so also can the line OC if its end 
C is shown. This is useful when, as in Fig. 4, several 
determinations of a harmonic are to be shown on one 
dial. Figure 4 shows forty dial points [12], each repre- 
senting the determination of LZ. in Batavia pressure for 
one of the forty years 1866 to 1905. 
The Probable Error. More than a century after La- 
place’s pioneer attempt to assess the reliability of his 
determination of Li, [74c, d], Bartels [11, 12] in 1926 
applied the theory of errors in a plane to assess the 
uncertainty of the mean of a number of independent 
determinations of a periodic variation such as Ls, con- 
veniently represented by dial points as in Fig. 4. He 
showed how to determine the probable error ellipse cen- 
tred on the dial point C for the mean determination. 
When, as in Fig. 4, the individual dial points are sym- 
metrically distributed around C, the ellipse becomes the 
probable error circle, as there shown. Its radius r is cal- 
culated, according to the theory of errors, from the 
distances d of the individual dial points from C. When, 
as in Fig. 4, the individual dial points are of equal 
5. For the solar semidiurnal component variation of baro- 
metric pressure at Washington, D. C.; Fig. 3a is the graph of 
$2 sin (2t + G2). 
DYNAMICS OF THE ATMOSPHERE 
weight, r = 0.939d, where d denotes the mean value 
of d. 
The probability that any individual dial point will 
fall within a distance d from C is 1 — e¢ “/””, where 
x = 0.833d. When d =r, this probability is 44, sig- 
nifying that half the points are likely to fall within the 
probable error circle. This chance is reasonably well 
exemplified in Fig. 4, where nineteen of the forty points 
lie within the circle. 
Fic.4.—Harmonic dial showing the forty dial points for 
determinations of Z»., the lunar semidiurnal air tide in Batavia 
barometric pressure, for each of the years 1866-1905. Their 
centroid is the point C, specifying the forty-year mean of Lp. 
The circle with C as centre is the probable-error circle for any 
of the forty yearly dial points. 
The circle indicates the probable error r of each of 
the forty yearly determinations of Z2; that of the 40- 
year mean is r/+1/40, as might be indicated on a sep- 
arate dial showing only C and its own probable error 
circle. 
A determination may be considered reasonably good 
if the length of its dial vector is at least three times its 
probable error. For LZ, at Batavia, each yearly deter- 
mination satisfies this criterion, as there r is 0.011 mm, 
and J, ranges from about 0.045 to 0.078; for the 40- 
year mean value, J. = 0.062 and r = 0.0016, so that 
1s/ r= 39. 
The uncertainty in the amplitude of a harmonic vari- 
ation may be regarded as corresponding, in the sense 
explained above, to its 7, and that of the phase to the 
angle sin! (r/l) subtended at O by half the probable 
error circle. Thus the phase uncertainty will be greater, 
for a given value of r, the smaller the amplitude J; for 
the ‘‘yearly” probable error circle shown in Fig. 4, the 
phase uncertainty is 10°, or +20 minutes of (lunar) 
time; for the 40-year mean it is only three minutes of 
time. 
Methods of Computation of Z.. Several different 
methods have been used to determine L, from baro- 
metric records at various stations. In a few cases lunar 
hourly readings were taken, or measured from baro- 
graphs, or interpolated between the solar hourly meas- 
ures. This is quite unnecessary; the solar hourly values 
fully suffice if properly used. Where only a few readings 
