BC8F. cosec. 2d, in seconds. This is usually called the 8F2.co fourth correction, and is the same as the fourth correction of Witchell's method, which is generally neglected. The last term MS — MC is equal to 4. SC2. cot. d, or E. cot. d, nearly. This is found in Table H, the argument at the side being E, and at the top d. It is called the fifth correction, and within the limits of the present table it is additive like the other corrections, though in fact, if it were of any sensible magnitude, it would be subtractive when the distance is above 90°, but for such distances and for altitudes exceeding 5° this correction is only a fraction of a second, and it is in all cases neglected by Witchell and Lyons. Find the app. dist. at the top, and F at the side, the corresponding number is the fourth correction additive. 18901880186 1840182°180° 178° 175° 172° 170° 165° 160° 55°150°145 19101920194°196°198° | 100° | 102° | 105°|108°|110°|115°|120°| 01 01 01 0′′| 0′′| 0′′| 0 0' Table H. 10 11 = Thus in the preceding example, we have E 1', F = 2', hence the fourth correction in Tabie G is 1". The fifth correction in Table H is less than a second. Both these corrections are insensible; and this is generally the case except the distance of the objects is very small, and the altitudes very low; but observations are rarely made in such circumstances, on account of the uncertainty of the refraction near the horizon, which is a much greater source of error than the corrections now under consideration. For a variation of ten degrees in Fahrenheit's thermometer would produce an alteration of 15" in the refraction of a body, situated 5° above the horizon; this might produce a correction of the same order in the true distance, which would in general be much greater than the sum of the fourth and fifth corrections. Now as Navigators do not usually notice the corrections depending upon the thermometer and barometer, it becomes necessary to avoid those observations, in which the corrections for temperature and density would be great, or in other words, the low altitudes. But if the objects are sufficiently elevated to render the corrections for the temperature and density small, the fourth and fifth corrections will be hardly sensible, so that for all practical purposes, it will be sufficient to notice the rest of the corrections in the preceding formula (A) and neglect, the two last. In addition to this we may observe that the quantities thus neglected are not in general greater than those depending upon the spheroidal form of the earth, which are rarely, if ever, taken into consideration; neither are they greater than the errors to which the lunar tables are liable, and it appears to be an unnecessary degree of accuracy to notice equations which are within the limits of the errors of those tables. V. On the method of computing the Dip of the Magnetic Needle in different latitudes, according to the theory of Mr. Biot. AN BY NATHANIEL BOWDITCH, LL. D. An article upon the "Variations of the Terrestrial Magnetism in different latitudes," published in vol. 22 of Tilloch's Philosophical Magazine for the year 1805, contains the formulas discovered by Messrs. Biot and Humbolt, which represent with a considerable degree of accuracy all the observations of the Dip of the Magnetic Needle that have been made in various parts of the world, particularly in the northern hemisphere. The same subject is continued in another paper in the 49th volume of the same work for the year 1817, in which there is a small alteration in the formula; but the method may be yet considerably simplified, as will be shown in the present paper, in which I shall briefly mention the principles of Mr. Biot's theory, his formulas, and the improvement of which those formulas are susceptible. According to this theory there are two magnetic poles P, p, upon the earth's surface, the northern in the latitude of 78° N, and in the longitude of 25° W. from Paris nearly; the southern in 78° S. and 155° E. nearly. The line P p joining these two poles is called the magnetic axis. The plane PEP Р P N B passing through this axis, and any proposed place E upon the earth's surface, is the magnetic meridian of that place. The plane passing through the earth's centre C, perpendicular to the magnetic axis, is the magnetic equator. The magnetic latitude is counted from this equator, and is equal to the complement of the angle PCE. Upon the magnetic equator the dip is nothing or very small, and at the magnetic poles is nearly equal to 90°. In any other place the dip is computed upon the principle that the whole magnetic force of the earth is concentrated in two magnetic points S, N, situated in the magnetic axis P p, at infinitely small equal distances from the earth's centre C. The forces of both these magnetic points are supposed to be equal to each other; the one being attractive, the other repulsive, both of them varying inversely as the square of their distances from any attracted point E of the surface. = u = 90° Suppose the earth to be of a spherical form, and let its radius CP=CE = 1, also CS = CN=a, the magnetic latitude of the point E equal to a, and the angle PCE -A. Then drawing the line s Cn perpendicular to EC, we shall have nearly Ss = N n = x. sin. a; C8 Cnx. cos. ^, and as x is supposed to be infinitely small we shall have, by neglecting x2, x3, &c. EN-CE-Nn 1-x. sin a; ES = = CE+Ss1+x. sin a; and if the magnetic force of either of the points S, N, upon a place at the distance 1 is F, the force of the point N upon E will be by hypothesis F = F. (1+2. x. sin. a) in the direction EN. This may be reduced to two other forces; the one in the vertical direction E C, which will be represented very nearly by F (1+2x. sin. ); the other in the horizontal direction Cn equal to F.x. cos.λ. In like manner the repulsive force of the point S will be -2x sin. λ) in the direction S E, which represented by F === F (1 |