In this case da = 56959, b = 19695, (see Art. 135) ; .. λ = 11° 27′ 33′′ = 0·2 in arc, cos 2m = cos 47° 34′ = 0·6747; .. SR = −0·0025 (Ɛa + db) + 0·0050 (da — db) = 0·0025 Sa - 0.0075 Sb = 5 feet. Although the ellipse compared with the mean ellipse differs much in the length of its axes, yet its depression at the middle point of an arc eleven degrees long, is only 5 feet. PROP. Geodesy furnishes no evidence, in proof or disproof, of the upheaval or depression of the Earth's surface as suggested by geological phenomena. 156. It might at first seem from the last Article that geodesy proves, that the position of the arc has not been sensibly changed, and that geological processes have not affected it. But it must be observed, that the comparison of the arc has been made not with the mean ellipse itself, but with an ellipse equal in dimensions to the mean ellipse and with axes parallel (because the latitudes are measured in all the ellipses from the same or parallel lines). This ellipse was so drawn as to pass through the extremities of the arc; but we have no means of knowing that the mean ellipse itself passes through those two points: it may lie above them or below them. We have no means of ascertaining the precise position of the centre of the mean ellipse. The only way of doing this is to make a geodetic measurement of the whole of one meridian from pole to pole. Till this is done we have no evidence of any particular arc lying above or below the mean, i.e. of its having been elevated or depressed. The greatest geological changes of level, therefore, are perfectly consistent with all we know by geodesy of the surface of the Earth. 157. It has been explained, that in consequence of the inequalities of the Earth's surface the observations, whether made on the pendulum or in geodetic operations, are all referred to the SEA-LEVEL; that is, to that surface which the sea would form if allowed to percolate by canals through EFFECT ON SEA-LEVEL. 155 the continents. The sea is thus taken as the basis of our measurements; and is generally assumed to have a spheroidal form. But it is possible that these local disturbing forces, arising from attraction, may have the effect of crowding up the waters in the direction in which the forces act, so as sensibly to alter the sea-level from the spheroidal form. This we shall proceed to examine. PROP. To find the effect of a small horizontal disturbing force in changing the Level of the Sea. 158. Let U be the disturbing force and du an element of the line u along which it acts. Then (Udu must be added to the potential in the equation of fluid equilibrium of Art. 75. 2 V + 20° r2 (1 − μ3) + (Udu = const. at the surface. When the small force U is neglected, ar=1+e.μ2, by the equation to the ellipse. Hence, neglecting small quantities of the second order, dividing by E, multiplying by a, and transposing, the above equation must become 1 dr Now r de = is the tangent of the angle between r and the normal, tan suppose: and the angle through which the normal is thrown back by the force U Hence the element ds of the undisturbed meridian line on the surface of the sea is elevated, on the side towards which It has been shown (Art. 136) that although there are causes (such as the Himmalayas and the Ocean) which produce a considerable amount of local attraction, yet that on the whole they very nearly balance each other. The following three examples are therefore solely for applying the formula. 159. Ex. 1. The Himmalayas attract places along the coast of Hindostan with a force varying nearly inversely as the distance from a line running E.S.E. and W.N.W. through a point in latitude 33" and longitude 77° 42', and equal to g tan 7" at 1020 miles distance: (see Phil. Trans. 1855, p. 91, 94; also 1859, p. 793). Find the effect this would have upon the sea-level between Cape Comorin and Karachi, which are about 1600 and 775 miles from this line, if there were no counteracting cause, as it is believed there is (Art. 136). In this case Ug tan 7" (1020÷u) u is the distance from the line. We may take the arc for the chord. fore rise of sea-level from this cause There Ex. 2. As the distance from the line increases the force will vary more as the inverse square. Suppose that to the distance 1020 miles it varies as the inverse distance, and beyond that as the inverse square. For the first we must integrate as above: thus GREAT BRITAIN AND NEW ZEALAND. 157 For the more southern part U=-g tan 7" (1020÷u), and the rise of the level The sum of these is 116 feet, and is somewhat less than the result before obtained. We shall not be above the mark, therefore, in taking the latter. Ex. 3. If u be the distance, in linear degrees, of the parallel of any place on the west coast of Hindostan from that of Cape Comorin, then the force acting towards the north at any point of that coast, arising from the deficiency of matter in the Ocean, may be approximately represented by the following formula (see Art. 62, Ex. 2): (0·000095556839-0·000002836162 u+0·000000004072u) g. Hence at this place the sea-level would be higher than at Cape Comorin, in consequence of this cause, by 0.000095556839u – 0·000001418081u2 + 0·000000001357u3. Karachi is about 17° north of Cape Comorin. Hence from this cause, the sea would be higher at Karachi than at Cape Comorin by 0.00122 of a linear degree=0·8489 mile =448 feet, if there were no other cause in operation to counteract it. Ex. 4. To find how much higher the sea-level stands on the shores of Great Britain than it would, if the Ocean in the New Zealand hemisphere were to become land, all other things remaining as at present. If a great circle be drawn upon the earth as an equator, New Zealand and Great Britain (which are nearly in each other's antipodes) being its poles, the New Zealand hemisphere is nearly all water. We must find the effect of the deficiency of matter in this ocean hemisphere in producing horizontal local attraction in the opposite hemisphere. (1) We will suppose that this effect is the same as if the New Zealand ocean were of the form of a hemi-spheroidal meniscus, of thickness h at New Zealand: then by Art. 62, Ex. 3, the horizontal attraction at a place in the Great Britain hemisphere at a distance 6o from New Zealand (= W) = h p (0.1446 sin 0 + 0·0958 sin 20 + 0·0244 sin 30) 9, p being the deficiency of density in the ocean. Hence, supposing that this place is connected by a canal (as is the case in the North and South Atlantic Ocean) with the New Zealand hemisphere, the consequent elevation of the sea-level there is The density of sea water = 1.028: hence p.= 2·75 — 1·028 = 1·72, + 0.0479 cos 20+ 0.0081 cos 30) =0·626 h (0.1527 - 0·0958) at Great Britain = 0·626 × 0·0569 h = 0·036188 h mile =382 feet, if h = 2 miles. (2) Suppose that the ocean in the New Zealand hemisphere is considered to be of the form of a meniscus, the thickness at the pole being zero, and at the edge h. Then by Art. 62, Ex. 5, the elevation of the sea-level = −0·626 (2·0608 cos 0 + 0·9442 cos 20 + 0·2454 cos 30 +0.9442) h =0·626 (2·3C42 — 1·8884) h = 0·626 × 0·4158 h (3) Suppose that the ocean is regarded as uniformly deep. Then by Art. 63, Ex. 4, the elevation of the sea-level |