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IV. Side c = 90°, adj. angle A constant.

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A is of consequence constant.

III. Angle A and opp. side a constant.

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IV. Angle a and adj. side e constant.

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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

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b

Sa

tan c

-- becomes de: da :: tan c: a,

a

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and similarly of others.

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 Bс 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;

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

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2AC

sin 2B

consequently dac = dB

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 B 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

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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

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Hence when da - &c= O, which is the case when ac is a maximum, that is, when da = =d, sin 2a must =2c, and consequently 2a must be the supplement of 2c, or a + c must = 90%

Now, (chap. vi. art. 26) tanc = tan a cos B;

comp.

whence, rad: cos B :: tan a : tanc

and div.} rad + cos B: rad

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COS B:: tan a + tan c

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

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

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