Arc AR = 28° 57′ 3″ RK = - 0° 39′ 13″ Kq = - 1° 6′ 46″ Sum = 26° 51′ 4″ = corrected arc AQ. We have now the necessary elements in the Nautical Almanac, which we must reduce for the instant of the vortex passing the meridian in Greenwich time. July 2d. Meridian passage, local time, at 9h. 5m. A.M. " in Greenwich time 2d. 3h. 1m. Right ascension same time 56° 42′ 45″ Declination north " 18 00 1 Obliquity of the vortex " 26 2 32 Polar angle " 18 5 7 Arc AQ " 26 51 4 [Illustration: Fig. 14] PA = 17° 59′ 59″ } P = 128° 37′ 38″ PV = 26 2 32 } VA = 89 3 0 V = 47 59 44 VQ = 62 11 56 A = 20 3 42 PQ = 47 14 22 Q = 26 22 55 Latitude of Q on the sphere = 42° 45′ 38″ CORRECTION FOR PROTUBERANCE. 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 30° and 40°, 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.