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- 4":085 +2":1831 U+1":6203 V + x4,

0.403 + 4·1251 U+ 2.7756 V+x. The calculation makes U= -0.3856, V = 1.0620. The value of dr is zero when Z is substituted, as of course it

1 should be. Also a=20926348, b=20855233, e= Sir

294 Henry James takes as the final result for the mean figure of the Easth (see his Preface),

1 a=20926500, b=20855400, e=

294 The corrections for latitude in the example we have taken all along) are x=0":050, for Kalianpur -- 3":156, and for Kaliana 1":810. These are the quantities by which, according to the Principle of Least Squares, the observed latitudes must be altered to make the measured arcs accord with the mean ellipse above determined.

110. What has gone before leads to the determination of only the Mean Figure of the Earth. Any one meridian may possibly differ from this mean form owing to local causes, such as the rising or sinking of the surface from internal expansion and contraction of the materials of the crust, which may have taken place since the form ceased to be regulated by the laws of fluid equilibrium. Indeed the average result even seems to point out that some such change has occurred. For it appears in Art. 108, that the calculations in the Ordnance Survey Volume show, that there is a slight protuberance in the middle latitudes, even in the mean figure of the earth.

If there be any local deviations from the mean figure further than this, India seems to present phenomena which would suggest that these deviations must exist there if anywhere; and in that country an extensive and well-executed Survey has been carried on, which supplies us with data. The particular case of the Indian Arc has been used for illustration in the preceding references to the Ordnance Volume, because the formulæ will now be of use in the following calculations. In India there are visible sources of error in the position of the plumb-line which ought not to be overlooked. The mountain-mass on the North, and the ocean on the South by its deficiency of matter, both tend to give the plumb-line a deviation northward and through different angles. The author has approximated to the amount of these deflections, as before stated in this Treatise. The following are the results he has obtained. (See Phil. Trans. 1859.)

Deflections at Damargida, Kalianpur and Kaliana are :Caused by the Mountains 6":79 12":05 27":98

Ocean 10 44 9.00 6 -18

Totals 17 •23 21 05 34 •16 Errors in the amplitudes

3":82 ... 13" 11. If these be applied as corrections to the amplitudes in Ex. 1 of Art. 99, we have 1 = 5o. 23'. 37" + 13"=5o. 23'. 50", X= 60.3.56" +4" = 60.4'.0", and the formula of Art. 98 will give

1 a=20906792, b=20843795, e=

332' which is nearer the mean ellipse than the uncorrected data in Ex. 1 make it.

111. Captain Clarke has suggested the following course. By the principle of least squares he finds the ellipse which differs least from the mean ellipse in form, and gives deflections of the normal from the normal of the mean ellipse most in accordance with the calculated deflections. This he has done, taking account of mountain attraction only; the effect of the ocean on the plumb-line had not then been estimated. We propose now to go through his calculation, taking account of both these visible causes of disturbance.

PROP. To determine the ellipse which most nearly accords with the mean ellipse in form, and at the same time most nearly meets the anomalies in India arising from mountain and ocean attraction,

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112. Let ly, lg, lg be the latitudes of Damargida, Kalianpur, and Kaliana, corrected to the mean ellipse, so that (see Art. 109) the observed latitudes are l, -0':05, 1, + 3".16, and 18–1":81. If, now, taking a general case, la +ex, le+ég, 1x + eg are the latitudes for the three places referred to any other ellipse, then e, +0:05, 6, -- 3.16, e. - 1.81 are the corrections which must be added to the observed latitudes to make them accord with the new ellipse. Hence by Art. 109,

e, +0·05 = xy,
e, -3.16 =- 4:085 + 2.1831 U+1.6203 V + x,,

0.403 +4:1251 U + 2.7756 V+x,.
.. U=-0.3856 +1.8500e, – 4.4446e, + 2.5946eg,

V= 1.0620 – 3.10982, +6.6056e, – 3.4958ez. Suppose that dy, dg, dg are the angles of deflection caused by the mountains and the ocean. Then the ellipse which will most nearly satisfy the Indian Arc is that which makes

(e-d) + (e, - d) + (es - d.) +(1.8500e, -- 4:4446e,+ 2.5946e3)

+(-3.1098, +6.6056e, -3.4958e) a minimum. By differentiation, with respect to e, eg, es, we obtain three equations, which after transformation become

0.82493d, +0:30087d, -0.12583d,, e,= 0•30086d, +0.341990, +0.35716ds,

en=-0.12584d, +0.35715d, +0.76873d,, and from these we find

U=-0.3856 – 0·13760d, – 0.03671d, +0.17432d3,

V= 1•0620 – 0.13808d, +0.07484d2 +0.06325dg. The values of d,, dg, d, are now to be substituted: they are 17"-23, 21":05, 34":16; and they make e, = 16"-25, e,= 24":58, e,= 31":61; also U = 2.4255, V=2.4189.

2

e =

Hence the errors in the observed latitudes as affected by deflection, or e, +0":05, e-3":16, e, +1"-81, are 16":30, 21"-42, 33":42. These are less than the deflections by the small quantities 0":93, -0":37, 0":74. The values of U and V give the following results :

1 a=20919988, b=20846981, e=

287

113. We have thus obtained three different measures of the arc in question : viz. I. That derived from a comparison of the two portions of the arc together, the amplitudes not being corrected for local attraction; II. The same comparison after the amplitudes are corrected for mountain and ocean attraction; III. The ellipse obtained by least squares, which departs least from the mean ellipse in form and at the same time gives deviations of its normals from the normals of the mean ellipse as nearly as possible equal to the calculated deflections arising from local attraction. The results are here gathered together :

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Let da and 86 be the excess of a and b for each of these ellipses compared with the mean.

Hence

Arc 1.

Arc II. da = 10.93 miles, - 3.73, 86 = 3.77

- 2:20,

Arc III. -1.23, -1.60.

PROP. To find the difference in length, and also the distance at the middle latitude, of two elliptic arcs of small ellipticity, lying in the plane of the meridian, and having their extremities in the same points; the latitudes of those points being known approximately, and the ellipses to which the arcs belong having their axes parallel.

114. Let a and ß be the co-ordinates to the centre of the ellipse, of which a, b, e are the semi-axes and ellipticity, measured from some fixed point near the centres of the two ellipses. The squares and products of a - b, e, a and ß will be neglected. Let s be the length of the elliptic arc between the stations, l and l' the observed (or approximate) latitudes of the extremities, 2 and m the amplitude and middle latitude.

First. We will find the length of the arc. Let c be the chord, r and 0, r' and o' the polar co-ordinates from the centre of the ellipse to the extremities of the arc,

.. (=p+po?? - 2rr' cos (0-0')=2rr"{1-cos (0 - 0')}+(r-r')',

pra (1 – e sin’1), m= a (1 – e sino?').

Also

tan 0 =(1 – 2e) tanl, A=1- € sin 21;

.: 0-0=1- 2e sin cos 2m;

.. 1- cos (0–0')=1 - cos 2 – 2e sin', cos 2m

=2 sinon{1 – 2e (1 + cos a) cos 2m} ; :: d=4ao sino {1 – 2e (1 +cos ) cos 2m – € (sinol + sin*l')} = 4a" sino } [1 –e{1+(2+cos 2) cos 2m}];

sin n = 1+{1+(2 + cos 2) cos 2m}];

... sin

P.A.

*

* 8

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