transit instrument. It may be used like the zenith sector, for determining latitude by a star nearly on the meridian, the instrument having a spirit level parallel to the plane of the vertical circle. There are five vertical and five horizontal wires, the latter of which are convenient for observing single altitudes, or equal altitudes of the sun for either time or latitude. The mean of the times of the two limbs passing the five horizontal wires is taken as the time of the altitude shown on the vertical limb. A very accurate mode of determining the true time and error of a time keeper is by equal altitudes, morning and afternoon. If the sun did not change his declination in the interval between the two observations, half the interval in time added to the time of the morning observation would express the hour by the time keeper, when the sun was on the meridian. But as the declination does change, a correction of the half interval must be made, the formula for which is The correction is + if the declination is increasing, and if decreasing. The time of the sun's being on the meridian being corrected for the equation of time will give the time of mean noon by the watch, which will show the error of the watch. If a star be used instead of the sun, no correction is requisite for change of declination, the mean between the two times of observation must be compared with the computed mean time at which the star culminates, in order to have the error of the time keeper. If the readings be taken on the horizontal circle at the two times of observation, This is for error in the divisions of the limb, tested by running the microscope over them. CONVERSION OF ASTRONOMIC AND GEOCENTRIC LATITUDES. 365 the reading midway will correspond to the direction of the meridian. This also, in the case of the sun, requires a correction for change of declination in the interval, the formula for which is CONVERSION OF ASTRONOMIC AND GEOCENTRIC LATITUDE. From the nature of the astronomic instruments, the zenith point being determined by a plumb line, basin of mercury, or spirit level, it follows, by the law of gravitation, that the line from the station to the zenith is a normal to the elliptical meridian, and the angle which this line makes with the major axis or equatorial diameter will be the astronomic latitude, or latitude deduced from observation. This will be the angle MBA, in the diagram. and the formula for converting astronomic into geocentric latitude will be In which tan '9934 tan Aastronomic latitude, λ=geocentric latitude. RADIUS OF CURVATURE IN TERMS OF THE LATITUDE. We have had occasion to use an expression for the radius of curvature of the meridian considered as an ellipse, in terms of the latitude of the point of the meridian, under consideration, in various places in the part of this volume devoted to Geodesy. The following is the mode of deriving it. The ordinary expression for the radius of curvature of the ellipse found in elementary mathematical works is in which a and b denote the semi-axes of the ellipse x and y, tae co-ordinates of the point at which is the radius of curvature. If A denote the latitude of this point, since it is the angle which the normal makes with the major axis, we have since the part of the numerator in parenthesis is, by the equation of the ellipse, equal to a b2. From the last equation we obtain Substituting the second member of the last result for the numerator of (1) that formula becomes DETERMINATION OF THE FIGURE AND DIMENSIONS OF THE EARTH FROM THE MEASUREMENT OF TWO DEGREES OF THE MERIDIAN AT TWO DISTANT LATITUDES. The deviation of an oblate spheroid from a sphere is expressed by what is termed * The radius of curvature of a curve at any point is the radius of a circle having the same curvature as the curve at that point. its compression or oblateness. This is the ratio of the difference between its axis to the major axis in symbols, w representing the oblateness, omitting w2, which is a very small fraction, in consequence of the smallness of w, we may write But from (4) p. 366, applying either the binominal or McClaurin's theorem to the second member, we have p=a (1 e2 + 1⁄2 e2 sin2 λ + terms too small to affect the result) (3) By formula (5) p. 100, cos 2λ = 1-2 sino λ, substituting the second member of this last in place of the first in (3), that equation becomes p=a (1 — e2. e2 cos 2x + &c.) or by (2), p=a (1 — } w-w cos 2X) (4) If now & denote the length of a degree measured in the latitude λ, since P radius of the arc of the meridian in that latitude, is the If ' denote the length of a degree measured in another latitude λ', in a similar Performing the division in the second member, and neglecting the squares and higher powers of w we have 8 = 1+; w (cos 2λ' — cos 2λ) 1. Form a fraction which shall be the ratio of a degree in one latitude to its excess over a degree in another latitude. 2. Form another fraction, which shall be the ratio of unity to the difference between the cosines of the doubles of each latitude. 3. Take of the product of these two fractions. The value of a being known, that of a may be found from (5). W EXAMPLE. The length of a degree at the equator is ¿' 56.753 toises. in which N denoted the y normal, and the latitude. This formula may be deduced as follows:-The expression for the y normal in the ellipse is y normal = ( a1 y2+ b1 x2 A transit instrument, mounted in the prime vertical, or at right angles to the meridian, affords a very accurate method of determining latitude or declination when either is known, by observing the times of transit of the same star over the prime vertical on both sides of the zenith. Stars near the zenith are the best for this purpose, as the interval between the two transits is shortest for them, and there is less opportunity for instrumental changes between the two observations. The triangle PZS, which we have so often used in the preceding pages, which must, for our present purpose, be considered right angled at z, and in which the half interval between |