17. In the preceding table the differential relations are exhibited in equations, both because in that form they occupy comparatively small space, and because they are in the majority of cases more commodious in application than when presented after the manner of analogies. They may, however, at once be read and regarded as analogies, by taking the numerator and denominator of the fraction in each member of the equation, as the antecedent and consequent of a ratio. Thus, = cos c, becomes da: db:: cos c: rad

18. It must be borne in mind that the equations in this table are merely approximative, and that the results they furnish are more accurate in proportion as the variations are small. In cases where the greatest possible correctness is required, the theorems for the finite differences of sines, tangents, &c. must be introduced into the appropriate formula for the individual problem, instead of those for the differentials; and the results employed, instead of those tabulated in the preceding pages.

We here present a few examples of the use of the formulæ above given.

Example I.

Let an object whose height is AC, be measured by taking its angle of elevation ABC, at a given horizontal distance BC from its base. It is required to ascertain what error may be committed with respect to the height AC, in consequence of any supposed error in the observed angle B?

The fourth equation, class 4 of plane triangles, is applicable to this example:

But (chap. ii. 2, iv. 18), sin 2B = 2 sin B COS B;

This thrown into a proportion becomes,
As the sine of twice the observed angle,
To double the computed height;
So is the error of the observed angle,

To the corresponding error in height.

Corollary. Hence dAc, the error in the height will be a minimum, when sin 2B is a maximum, or is equal to radius. That is, the error in altitude, cæteris paribus, is the least when the angle of observation в is 45°.

Remark. If an error of a minute is made in taking the angle B, supposed of 45°, the corresponding error in the altitude will be its 1719th part: for in that case

Given the inclination of the plane of a theodolite to the horizon; required the greatest error that can possibly happen in determining the magnitude of an angle.

The face of the theodolite when posited horizontally, and when inclined in any assigned angle, may be regarded as two great circles of a sphere whose centre is the centre of the face of the theodolite. Also, if the

*The figure, as in many other cases, is left designedly to be sketched by the student,

index of the theodolite is horizontal when it passes through the points 0° and 180°; when it is turned round to measure any horizontal angle upon the inclined plane, a vertical circle passing through the index in that position, would have cut the theodolite, if posited horizontally, in the true arc. Hence (sketching a right angled spherical triangle like that at p. 87) if B be the angle of inclination of the theodolite, BC or a the arc measuring the angle upon the inclined face of the instrument, then will BA or c be the arc that measures the true angle; and it is required to determine in what circumstances a c is a maximum.

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Now equa. 5, class 2, of right angled spherical triangles, is applicable to the present case, and when adapted to our figure becomes

Hence when da

бо

sin 2c

0, which is the case when a -c is a maximum, that is, when dade, sin 2a must =2c, and consequently 2a must be the supplement of 2c, or a + c must = 90°.

Now, (chap. vi. art. 26) tan c = tan a cos B; whence, rad: cos B :: tan a : tan.c comp.rad+cos B : rad COS B:: tan a + tan c

and div.

and (chap. ii. art. 18) :: sin (a + c) : sin (a that is, rad + cos B: rad

COS B:: rad: sin (a — c). From this proportion, since в is given, the measures a, c, and their difference, may be determined.

But the expression admits of farther simplification.. For, transformed into an equation, it becomes.

that is, the tangent of half the angle of inclination is a

mean proportional between radius, and the sine of the maximum error on the theodolite.

Suppose that the face of the theodolite, instead of being horizontal, was inclined in an angle of 5 degrees. Then log tan B.... 2° 30'.... 8-6400931 Multiplied by..

2

To find when that part of the equation of time which depends on the obliquity of the ecliptic, is the greatest possible.

Here the sun's longitude will form the hypothenuse of a right angled spherical triangle, his right ascension will be the base, and the obliquity of the ecliptic is supposed constant. It is required to find when hyp. - base is a maximum; which is, evidently, as in the preceding example, when hyp. + base = 90°, that is, when sun's long.+ sun's right ascension = 90°, from the equinoctial points.

This happens in the year 1816, about May 7 and November 8.

Example IV.

To ascertain the error that may be committed in the observation of zenith distances, by any conceivable deviation of the instrument from the vertical plane.

Draw a figure in which HO is the intersection of the horizon with the vertical plane HZо, and нz'sо the position of the plane of the instrument, z being the apparent zenith upon that plane, and s the place of the heavenly body when it has arrived at the plane of the instrument. The arc zz' will measure the inclination I of the plane at H or o, and the base z's z of the right angled spherical triangle zz's will be the apparent zenith distance of the body, while the hypothenuse zsz+3 will be the true distance.

=

Now from equa. 6, art. 25, chap. vi. we have

cos zs = cos zz' cos zs: that is,

Cos I cos z = cos (z + x)

= COS Z COS X sin z sin x.

Whence sin z sin x = cos z (cos x

= cos z (1-2 sin2x-1+2sin21) 2 cos z (sin2 1 — sin2 x).

Dividing by sin z, there results

sin x 2 cot z (sin2 1 — sin2 1x).

Here since x is by hypothesis small, sin2 x is extremely minute, and may be rejected: hence sin x = 2 cot z sin2 1.

Corol. As the zenith distance z increases, the error æ diminishes, the inclination I remaining the same.

Supposing the deviation from the vertical 5', and the apparent zenith distance of a star 37°, the error would be2% of a second.

Note. If z, z', be the observed zenith distances of a circumpolar star (corrected for refraction) at the superior and inferior transits of the meridian; z, z', the true zenith distances; x, x', the corrections due to the deviation of the circle from the meridian: then will

x = z + x

x′ = z' + x' ;

therefore the distance of the pole from the zenith, or, (z) = (z′ + x' ́) — § (z + x). But in the inferior transit x is imperceptible; there

fore

lat. 90° (≈′ — ≈) = 90° — § (z′ — z) + 1x. Consequently, in the case of employing a circumpolar star, the latitude is only affected by half the error produced by the deviation of the instrument from the vertical plane.

Example V.

It is required to determine, at a given time and latitude, how long an interval is taken by the body of the sun to rise from the horizon.