E A 11 T H -5- du du du dx dy dz and if TT - < be the angle between this normal and the minor axis, so that </> is the latitude of P, we have Hence the equation to a " parallel " in which the latitude d> is constant is So that in an ellipsoidal earth the parallel is no longer a plane curve. Let longitude be reckoned from the plane of xz. As there are two species of latitude, astronomical and geoeentric, so there are in the ellipsoidal earth two species of longitude, geocentric (called w) and astronomical (called to). Conceive a line passing through the origin in the plane of the equator and directed to a point whose longitude is ^TT + W. The direction cosines of that line are sin o>, cos o), and 0. Those points of the surfacs whose normals are at right angles to this line are in the meridian whose longitude is w ; the condition of perpendicularity is expressed by x sin <a y cos w _ ~^~~ ~W~ and this, in fact, is the equation of the meridian, which is still on the ellipsoidal hypothesis a plane curve. The geocentric and astronomical longitudes are connected by the relation 2 tan -J 2 tan . This meridian curve is an ellipse whose minor semi-axis is c, and of which the semi-axis major is some quantity r intermediate "between a and I, such that 1 _ cos 2 u sin 2 u lake two quantities i, k, such that a 2 (l -i) = Jc- = r-i cos 2w) ; and take n such that =& 2 , then r + c and substitute the value of r, neglecting the square of i ; this gives . k + c 4 Now we have to determine not only the three semi-axes a, I, c, but the longitude of a. Let t^ be the longitude of one of the measured meridian arcs, u the longitude of a, then, for that arc, Jc c i n/ - + 7 cos2(w 1 -MO) k- c = +p cos 2% + q sm 2u lt where p i cos 2w , 4g f :=z sin 2tt . The normal at P does not pass through the axis of rotation, so that the observed latitudes ou an ellipsoid are not exactly the quantities which should be used in the ordinary method of ex pressing the length of a meridian arc in terms of the latitudes. But it may be shown that this consideration may be neglected. The data we have collected form 35 equations between the 40 ic-corrections to the observed latitudes, and the four unknown quantities determining the elements of the ellipsoid. Suppose x to be an approximate value of the ratio Jc - c : k + c, so that k- c where r is a small correction to x and suppose c x to be an approximate value of c so that c = c 1 (l+t), then the four unknown quantities are p, q, r, t. The result of making the sum of the squares of the 40 corrections a minimum is Feet. Metres. a- 20926350 = 6378294-0 fc= 20919972 = 6376350-4 c =20853429 - 63560681 a-c 1 "7" a-b 6-c_ 1 ~c ~" 313-38 _ ___ c ~ 3269-5 Longitude of a ............. 15 34 East. The meridian of the greater axis passes, in the Eastern Hemisphere, through Spitzbergen, the Straits of Messina, Lake Chad in North Africa, and along the west coast of South Africa, nearly corresponding to the meridian which passes over the greatest quantity of land in that hemi sphere. In the Western Hemisphere it passes through Behring s Straits and through the centre of the Pacific Ocean. The meridian (105 34 E.) of the minor axis of the equator passes near North-east Cape on the Arctic Sea, through Tong-kiug and the Straits of Suuda, and corresponds nearly to the meridian which passes over the greatest amount of land in Asia ; and in the Western Hemisphere it passes through Smith Sound, the west of Labrador, Montreal, between Cuba and Hayti, and along the west coast of South America, nearly coinciding with the meridian that passes over the greatest amount of land in that hemisphere. The length of the meridian quadrant passing through Paris, in the ellipsoidal figure given above, is 1000 147 2 5 metres, showing that the length of the ideal French standard is considerably in error as representing the ten-millionth part of the quadrant. The minimum quadrant, in longitude 105 34 , has a length of 10000024-5 metres. The probable error of the longitude of the major axis of the equator given above is of course large, as much perhaps as 15. It has been objected to this figure of three unequal axes that it does not satisfy, in the proportions of the axes, the conditions brought out in Jacobi s theorem. Admitting this, it has to be noted, on the other hand, that Jacobi s theorem contemplates a homogeneous fluid, and this is certainly far from the actual condition of our globe, indeed the irregular distribution of continents and oceans suggests as possible a sensible divergence from a perfect surface of revolution. If we limit the figure to being an ellipsoid of revolution, we get rid in our equations of two unknown quantities, and the result may be expressed thus : Feet. Metres. = 20926062 = 6378206 4 c =20855121 = 6356503-8 c:a = 293 98:294-98. As might be expected, the sum of the squares of the 40 latitude corrections, viz., 153 99, is greater in this figure than in that of three axes, where it amounts to 138 -30. In the Indian arc the largest corrections are at Dodagoontah, + 3""87, and at Kalianpur, -3" 68. In the Russian arc the largest corrections are + 3" 76, at Tornea, and - 3" - 31, at Staro Nekrassowka. Of the whole 40 corrections, 16 are under 1" 0, 10 between 1" and 2".0, 10 between 2" and 3"-0, and 4 over 3" 0. For the ellipsoidal figure the probable error of an observed latitude is 1""42 ; for the spheroidal it would be very slightly larger. This quantity may be taken therefore as approximately the probable amount of local deflection. In 1860, the Russian Government, at the instance of M. Otto Struve, imperial astronomer at St Petersburg, invited the co-operation of the Governments of Prussia, Belgium, France, and England, to the important end of connecting their respective triangulations so as to form a continuous chain under the parallel of 52 from the island of Yalentia on the south-west coast of Ireland, in longitude 10 20 40" W., to Orsk on the river Ural in Russia. This grand
undertaking was at once sst in action, but up to the present