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latitude and longitude between the stations, and the azimuth at the other station may be conveniently found.

Thus, as above, for the difference of latitude, let P be the pole of the earth, a a station whose latitude has been observed, and PAB the observed azimuth of B; it is required to find the side PB, and subsequently the angles at P and B. In the triangle PAB we have (art. 60. (a), (b), or (c))

cos. PB

Now, let

cos. A sin. PA sin. AB + cos. PA cos. AB.

be the known latitude of A; dl the difference between the latitudes of A and B (= Am if Bm be part of a parallel of terrestrial latitude passing through B); then l+dl is the latitude of B, and the equation becomes

sin. (1+dl)

cos. A cos. 7 sin. AB + sin. l cos. AB,

or (Pl. Trigo., art. 32.)

sin. l cos. dl+cos. l sin. dl=cos. A cos. 7 sin. A B+ sin. l cos. A B, or again (Pl. Trigo., arts. 36, 35.)

sin. 7 (1-2 sin.2 dl)+2 cos. l cos. dl sin.

whence

dl=

cos. A cos. 7 sin. A B+sin. 7 (1−2 sin.2 AB);

2 sin. 7 sin.2 dl+2 cos. l cos. 1⁄2 dl sin. 1⁄2 dl =

or dividing by 2 cos. 7,

cos. A cos. 7 sin. A B-2 sin. 7 sin.2 AB,

- tan. 7 sin.2 dl + cos.

dl sin. dl =

1

(cos. A sin. AB - 2 tan. 7 sin.2 AB).

Let the second member be represented by p; also for-tan. 7 put q, and divide all the terms by cos. dl; then

q tan.2 1⁄2 dl + tan. 1⁄2 dl (

or

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2

= p(1 + tan.2 dl),

(q—p) tan.2 1 dl + tan. 1⁄2 dl = p.

Treating this as a quadratic equation, we obtain

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or developing the radical term by the binomial theorem,

1

tan. 1 dl =

{2 (1-p)p-2 (q−p)2 p2+ 2 (q-p)

4 (q-p)3 p3 — &c.};

or tan. dl = p-qp2 + (1 + 2q2) p3, rejecting the powers of p above the third.

Next, developing the arc dl in tangents (Pl. Trigo., art. 47.) as far as two terms, we have

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= p

or

qp2 + (1 + 2 q2) p3 - p3, rejecting as before,

= p − qp2 + 2 p3 (} + q2);

whence dl 2p-2qp2 +

But 2p

=

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2 qp2 = cos. A sin.2 AB AB tan.21,

p3 (1 + 3g2).

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tan. 7 + 2 cos. A sin. AB sin.2

and pcos. A sin.3 AB, rejecting the remaining terms, which contain powers of AB higher than the third.

Now (Pl. Trigo., art. 46.) sin. AB = AB — † A B3; therefore the first term in the value of 2p becomes A B cos. A — AB3 cos. A; also the equivalent of p3 may be put in the form AB3 cos. A cos. A. This being added to the second term just mentioned, the three terms are equivalent to AB COS. A AB3 sin. A cos. A.

Again, putting

of 2p becomes

2

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AB for its sine, the second term in the value

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and the first term in the AB2 cos.2 A tan. l.

terms being added together, produce the term

-AB2 sin. A tan. l.

These

Thus we obtain for dl or am (the required difference of latitude)

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AB3 sin. A cos. A (1 + 3 tan.2 7) &c. Here AB and dl are supposed to be expressed in terms of radius (= 1); if AB be given in seconds, we shall have, after dividing by sin. 1",

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dl (in seconds) = AB cos. A AB2 sín. 1" sin. A tan. 7AB3 sin. 1′′ sin.2 a cos. a (1 + 3 tan.2 7) · &c. This formula is to be used when the angle PAB is acute: if that angle be obtuse, as the angle PAB', its cosine being negative, the formula would become

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A B3 sin.2 1′′ sin.2 a cos. A (1 + 3 tan.2 1) &c.

If in either of these expressions we write for 7 the term

dl, in which case represents the latitude of B; then since, as a near approximation,

tan. (l'— dl) = tan. l' tan. dl,

the term AB2 sin. 1" sin.2 A tan. 7 becomes

A B2 sin. 1" sin.2 a tan. l' +

AB2 sin. 1′′ sin.2 a tan. dl;

and for dl putting AB sin. 1" cos. A, its approximate value, we get

AB2 sin. 1" sin.2 a tan. l' + AB3 sin.o 1′′ sin.2 a cos. a. If to the last of these terms we add the term A B3 sin.2 1′′ sin. A cos. A, the first expression for dl above becomes, neglecting the terms containing tan.2 7,

2

A B COS. A AB2 sin. 1" sin.o ▲ tan. l' +

AB3 sin.2 1" sin.2 a cos. A ;

in which all the terms are negative when the angle at a is obtuse.

This formula corresponds to that which denotes the value of Mp in art. 360.; and from it we should have

PA PB + AB COS. A

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AB2 sin. 1" sin.2 A tan. l'+ &c. conformably to the value of Pz in art. 352.

The angle at P, or the difference of longitude between the stations A and B, and also the azimuthal angle at B, will be most conveniently found by the direct formulæ III. and IV. above given.

Delambre observes that the meridional arcs computed on the hypothesis that the earth is a sphere do not differ sensibly from those which would be obtained on the spheroidal hypothesis; because the sums of the spheroidal angles of the triangles are, without any appreciable error, equal to those of the spherical angles, and the chords of the sides are the same. Thus we obtain correctly the sides of the triangles and the arcs between the parallels of latitude whatever be the figure of the earth (provided it do not differ much from a sphere). He adds that the consideration of the spheroidal figure of the earth is of little importance in the calculations, except for the purpose of changing into seconds the terrestrial arcs which have been measured or computed in feet. This may be done by dividing such arc by the radius of curvature at the place, instead of dividing it by what may be assumed as the mean radius of the earth.

P

a A

b

B

411. The length of a degree of latitude being found, from the results of admeasurement, to be greater near one of the poles of the earth than near the equator, it follows that the earth is compressed in the former region, or that the polar semi-axis is shorter than a semidiameter of the equator. For, let A a and вb represent the lengths of two such

N

N/ R/

E

terrestrial arcs, supposed to be on the same meridian, and

imagine AD, a D, BE, bE to be normals drawn to the earth's surface; the angles at D and E will each be equal to one degree and since Aa, Bb may be considered without sensible error as portions of circles, the sectors A Da, BEb will be similar, and the lines AD, BE, which may be considered as radii of curvature, will be to one another in the same ratio as the arès. Therefore Aa, which is the nearest to the pole P, being longer than Bb, AD will be longer than BE, or the surface of the earth about a will be less convex than the surface about B, which is the nearest to the equator.

412. Adopting now the hypothesis that the earth is a spheroid of revolution, such that all the terrestrial meridians are ellipses whose transverse axes are in the plane of the equator, and the minor axes coincident with that of the earth's rotation; the principal investigations relating to the values of terrestrial arcs will be contained in the following propositions.

PROP. I.

Every section of a spheroid of revolution, when made by a plane oblique to the equator, or to the axis of rotation, is an ellipse.

Let XYC, XPC and PCY be three rectangular co-ordinate planes formed by cutting a spheroid in the plane of the equator (XYC) and in those of two meridians at right angles to one another; and let a plane cut the spheroidal surface in AB and the plane X PC in AR, which for the present purpose may be supposed to be a normal to the spheroid at A; it is required to prove that the section is an ellipse.

A

F

E

D

L

R

Y

P

Let CD (), DE (= y) and EB (z) be rectangular co-ordinates of any point B in the curve line AB; and let RF(2), FB (y) be rectangular co-ordinates of B with respect to AR: also let x 7 represent the angle ARX, and the inclination of the normal or vertical plane ABR to the co-ordinate plane X PC. Imagine a perpendicular to be let fall from в on the plane XPC, it will meet the plane in &; and join G, F: draw also FL in the plane X PC perpendicular, and GK parallel to XC. Then we shall have, BF and FG being perpendicular to AR so that the angle BFG is the inclination of the plane ABR to the plane X PC,

y (= BG or ED, or BF sin. BFG) = y' sin. 0;

also, DL or GK being equal to GF sin. GFK or GF sin. 7, and GF to BF cos. BFG or BF cos. 0,

X = CD or CR + RL+DL) = CR + x' cos. l + y' cos. O sin. 7; Z = GD or FL — FK) = x' sin. 7 y' cos. O cos. l.

Substituting these values of x, y and z in the general equation Az2 + B (x2 + y2) = c of a spheroid, the latter will manifestly become of the form

mx2 + ny12 + px'y' + qx' + ry' = s,

which is the equation to a line of the second order: the variables in it being the assumed co-ordinates of B, it follows that the equation appertains to the curve line AB; and since the curve, being a plane section of a solid, returns into itself, it is an ellipse.

PROP. II.

413. When the sides of a terrestrial triangle do not much exceed in extent any of those which are formed in a geodetical survey, the excess of the three angles above two right angles is, without sensible error, the same whether the earth be considered as a spheroid or a sphere.

This may be proved by comparing the angles of a spheroidal, with those of a spherical triangle, when the angular points of both have the same latitudes and equal differences of longitude, agreeably to the method pursued by Mr. Dalby in the "Philosophical Transactions" for 1790.

Let PRM be a spheroidal triangle of which one point as P is the pole of the earth, and let prm be a corresponding triangle on the surface of a sphere: let also ABC be the plane of the equator. Imagine the normals MD and RE to be drawn cutting AC and BC in g and h; then the angles MgA RhB, which express the spheroidal latitudes of м and R, will be equal to the angles mca, rcb respectively; and we shall have MD parallel to mc and RE parallel to rc. Imagine

B

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M

h

a

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P

m

C

E

also the planes DMR, ERM' to pass through MD and RE parallel to the plane rmc; then R'D will be parallel to rc, and

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