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119. It will be observed, that the three arcs, which have been examined in Arts. 114-116, were compared, 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). For this ellipse was so drawn as to pass through the extremities of the arc; and we have no means of knowing that the mean ellipse passes through those two points. It may lie above them or below them. We have no means of ascertaining the 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.
120. In consequence of the inequalities of the Earth's surface, levelling 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 the continents. The sea is thus taken as the basis of our measurements; and is assumed to have a spheroidal form. But it is possible that 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.
121. Let U be the disturbing force and du an element of the line u along which it acts. Then Udu must be added to d V in the equation of fluid equilibrium of Art. 73.
me (1 —-?) + | Udu=const. at the surface.
Putting w'=m. E-ai and substituting for V (Art. 91);
m) Ea? /1 constant tle
| 2 73
When the small force U is neglected, a=r=1+€.M. Hence, neglecting small quantities of the second order, dividing by E, multiplying by a, and transposing,
1 dr Now
is the tangent of the angle between r and the normal, = tan f suppose: and the angle through which the normal is thrown back by the force U
a udu = df = 8. tant
do E d0 Hence the element ds of the undisturbed meridian line on the surface of the sea is elevated, on the side towards which U acts, by the space
:: whole elevation of the sea-level = "L
integrated between the limits.
122. 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 upon the sea-level between Cape Comorin and Karachi, which are about 1600 and 775 miles from this line, arising from this cause.
In this case U=-g tan 7" (1060 = u) u is the distance from the line. We may take the arc for the chord. Therefore rise of sea-level from this cause
0.3148 = 1020 tan 7" loge miles = 0·0346 x 775
0:414 = 0.025 mile = 132 feet.
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
0:1193 0.0346 loge = 0·0346 = 0·0095 mile = 50 feet. 775
For the more southern part U=-g tan 7" (1020 = u)', and the rise of the level
=0.0346 x =0.01254 mile=66 feet. 1600)
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 using 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 Phil. Trans. 1859) :
(0.000059556839 – 0.000002836162u + 0.000000004072u*) g.
Hence at this place the sea-level is higher than at Cape Comorin, in consequence of this cause, by
0.0000595568394 – 0.000001418081u+ 0.0000000013570%.
Karachi is about 17° north of Cape Comorin. Hence, from this cause, the sea is higher at Karachi than at Cape Comorin by 0.00122 of a linear degree = 0.8489 mile = 448 feet.
Ex. 4. Suppose an attracting force resides in Kalianpur, sufficient to produce a deflection 7". 75 at 400 miles' distance, and that the force varies inversely as the square of the distance; find its effect on the level between Cape Comorin and Karachi.
In this case the force is g tan 7" •75 (400 – u)", u being expressed in miles;
... rise in level = 160000 tan 7''.75
= 21 feet,
Uo u, and u, being 1125 and 643 miles, the distances of Kalianpur from Cape Comorin and Karachi.
Taking the sum of these three causes together, the increase in height of the sea-level at Karachi above that at Cape Comorin is 116 + 448 + 21 = 585 feet. There may be also other causes which may increase or decrease this result. But it serves to illustrate to what extent local attraction may
have an effect upon the standard level to which all heights are referred.
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