We have hitherto considered the earth a perfect sphere with a diameter
of 7,900 miles. It is convenient to regard it thus, and afterwards make
the correction for protuberance. We will now indicate the process for
obtaining this correction by the aid of the following diagram.
[Illustration: Fig. 15]
Let B bisect the chord ZZ'. Then, by geometry, the angle FQY is equal to
the angle BTF, and the protuberance FY is equal the sine of that angle,
making QF radius. This angle, made by the axis of the vortex and the
surface of the sphere, is commonly between 30d and 40d, according as the
moon is near her apogee or perigee; and the correction will be greatest
when the angle is least, as at the apogee. At the equator, the whole
protuberance of the earth is about 13 miles. Multiply this by the cosine
of the angle and divide by the sine, and we shall get the value of the
arc QY for the equator. For the smallest angle, when the correction is a
maximum, this correction will be about 20' of latitude at the equator;
for other latitudes it is diminished as the squares of the cosines of
the latitude. Then add this amount to the latitude EQ, equal the
latitude EY. This, however, is only correct when the axis of the vortex
is in the same plane as the axis of the earth; it is, therefore, subject
to a minus correction, which can be found by saying, as radius to cosine
of obliquity so is the correction to a fourth--the difference of these
corrections is the maximum minus correction, and needs reducing in the
ratio of radius to the cosine of the angle of the moon's distance from
the node; but as it can only amount to about 2' at a maximum under the
most favorable circumstances, it is not necessary to notice it. The
correction previously noticed is on the supposition that the earth is
like a sphere having TF for radius; as it is a spheroid, we must correct
again. From the evolute, draw the line SF, and parallel to it, draw TW;
then EW is the latitude of the point F on the surface of the spheroid.
This second correction is also a plus correction, subject to the same
error as the first on account of the obliquity, its maximum value for an
angle of 30d is about 6', and is greatest in latitude 45d; for other
latitudes, it is equal {6' x sin(double the lat.)}/R.
The three principal corrections for protuberance may be _estimated_ from
the following table, calculated for every 15d of latitude for an angle
of 30d, or when the correction is greatest.
Latitude. 1st Corr. 2d Corr. 3d Corr.
0 + 20' + 0 - 2
15 + 19 + 3 - 1.5
30 + 15 + 5 - 1.5
45 + 10 + 6 - 1.
60 + 5 + 5 - 1
70 + 1 + 3 - 0.5
We can now apply this correction to the latitude of the vortex just
found:
Latitude on the sphere 42d 45' 38" n.
Correction for protuberance + 14 22
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Correct latitude 43 00 00