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103. COR. The expansions of ac' and y' are as follows:

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PROP. To find an expression for measuring the departure of the curve of the meridian from an ellipse at any point, when the meridian is not elliptical. 104. Let

xy be the co-ordinates to any point of which l is the latitude, and the radius of curvature as above.

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(4 B) cosl+} (B-C) cos 31 +C cos 51,

1
y=(A + B) sinl+*(B+C) sin 31+ C sin 51.
y31+

Ž©sin 5

Let a and b be the semi-axes of the curve, whether it be an ellipse or not. Hence these values of w and y give

a=A – B+}(B-C)+c= AB-C
0 = 4+B-}(B+C)+{C= A+

A+B-C,
1-11 (1+3), neglecting B.C &c.

8 B

e=1

3 A

If we put these for a and e' in the expressions for ac' and y' in Art. 103, we shall have the co-ordinates of a point (latitude 1) in the ellipse constructed upon the same axes as those of the actual curve. After reduction, we get

2

1 B2 x'= A-B

5 B2

1 B cosl+ B.

cos 31+ cos 51, 15 9 A

18 A)

6 A

5 B? 18 A

sin 31+ĞA

1 B2

sin 5l; 6 A

3=(4+B-*c+) sin+({B:: x-x'=(C

(0-5) (1 cos 31 +{cos 51), y—y=(c- 6) (iš sin l+j sin 81 +} sin 52).

cos 7

1 3

Let 's and Sr be the distances between the points (wy) and (a'y') measured along the arc of the ellipse and the normal;

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PROP. To obtain a formula for correcting the amplitude of an arc, so as to make its measured length accord with a given

curve.

105. Let s be the length of the arc and p the radius of curvature as before; then, by integration,

8=Al+B sin 21+ 4C sin 4l+constant. Let 7-10,1+14 be the limits of s, 1 being the latitude of the middle point, and the amplitude of the arc;

.. s= A$+2B cos 21 sin $ + C cos 41 sin 20. Suppose now that xx,' are the small corrections which must be applied to the observed latitudes, l-14, 1+1$, to make them accord with the measured length 8.

Then 1-10+ and 1 + $ + , and $ + xy' – X, must be put instead of ?-14, 1+14, and $ in the above formula. Hence, neglecting the squares of small quantities, 8= A (+ x,' — «) + 2B cos 21 {sin $+ (x,' - x) cos ®}

C cos 47 {sin 26+2 (a' – x) cos 20};

:: (oc' - x) (A + 2B cos $ cos 27) =s - A° -2Bcos 2l sin -C cos 41 sin 20. Put

A + 2B cos o cos 21= A=;

2 Bu :: x'- x = M.

sin

21 A

A

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Cu sin 26 cos 41.

COS

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x'=m+AU+BV + y2 + xy, where m, a, b, y are functions of the observed latitudes, the measured length, and numerical quantities only.

106. As an example which the student may work out for himself, the following is selected from the Volume of the Ordnance Survey.

Observed

Station.

Measured Arcs in feet.

Latitudes.

Amplitudes.

Damargida 18°3' 15"-292
Kalianpur 24 7 11 •262 6° 3'55"-970
Kaliana 29 30 48 322 | 11 27 33 .030

22029047

4164042.7

If x, be the correction for Damargida, then the formula, when the numbers are substituted, will give these corrections : For Kalianpur...–4.063 + 2.1831 U+1.6212V + 0.4285Z+X,,

... Kaliana......+0.365 + 4.1251 U+ 2.7741 V – 0.7213Z+ x,. Also

dr - 29.01 -1•16 V - 557.07 Z feet.

PROP. To explain the process by which the mean figure of the earth is obtained from the observed latitudes and the measured arcs by the Principle of Least Squares.

107. Suppose that we have a number of equations with numerical coefficients connecting a number of unknown quantities, less in number than the equations; if the equations are true, the same values should come out whichever of the

equa, tions we use in the process of elimination. In Physical Science it often happens that we have a problem of this sort, in which the numerical coefficients, being obtained from observations, are not exact, but only approximate. If the correct values of the unknown quantities were substituted, they would not exactly satisfy the equations, but small residuary errors will appear, differing according to the set of equations we select for elimination. It would seem, therefore, difficult to determine which of all the results actually obtained is nearest the truth. The late Professor Gauss discovered the Principle of Least Squares, which is of eminent service in such cases of perplexity. The principle is this ; that those values of the unknown quantities are nearest the truth which make the sum of the squares of the errors the least possible. In using this principle the Differential Calculus will evidently furnish us with exactly as many equations as there are unknown quantities; and the problem will be solved, with the nearest approximation to the truth attainable.

In this manner the Mean Figure of the Earth may be determined. In the Volume of the Ordnance Survey eight arcs in Europe and India, consisting of 66 subordinate portions, have been used. In each arc the errors in the latitudes of the principal stations, which divide it into its subordinate portions, are calculated, as in the last Article, in terms of UVZ and the unknown error (as x) in one of the terminal stations, or in any one of the stations chosen as a starting point. The eighť arcs will thus furnish 66 formulæ of correction, (similar to that for a' in Article 105), involving eleven unknown quantities U, V, Z, X, X, ... Xs. The nearest values of these are obtained by differentiating the sum of the squares of these corrections with respect to these unknowns and equating the results to zero. The process of calculation is very

laborious.

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108. In the Ordnance Volume U=-0.6937, V=1.4838, Z=0.3739; and these make

1 a=20927197 feet, b=20855493, e=

292 and the value of dr in Art. 104 becomes 117.5 sina 21 feet, which shows that the greatest departure from the elliptic form is in latitude 45', and equals 117.5 feet. The correction of the latitude of Damargida (i.e, the value of x, in Art. 106) is – 0"-246, and the consequent corrections for Kalianpur and Kaliana are — 3":578 and i":643, for the above mean values of U, V and Z. The above measures determine that curve which is nearest to the meridian, of all the curves represented by the general formula in Art. 101. It appears to be very nearly elliptical, bulging out but slightly in the middle latitudes.

PROP. To explain the process of finding the Ellipse most nearly representing the observations.

109. The process is precisely similar to that explained in the last Articles, C being first made equal to 5B: 64, that the curve may be an ellipse. By Art. 105, Z 5 1 V

V

.. Z= 24 (200 10000/

96 480

5

(200

+

0000)

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10000

There will be only ten unknown quantities in this case. The corrections for Kalianpur and Kaliana in terms of x,, that for Damargida, are

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