The following example was likewise taken by an in strument made by Mr Troughton.
The angle thus measured requires to be corrected for the eccentricity of the lower telescope ; and this correc tion depends on the distance of the objects from the observer, and upon the distance of the axis of vision of the lower telescope from the centre of the instrument.
In circles constructed by Mr Troughton, the eccen tricity of the lower telescope is one inch and four tenths; and the following Table is calculated upon this supposi tion from Delambre's formula given in L'4rc du illeridien, In making the above observations the light was favour able, being steady and uniform ; the sun was always hid.
The mark exceeded the wire some seconds on each side, and was not so round as could have been wished. A mark smaller and perfectly round might probably have been bisected more exactly.
In the 24th observation the day light began to fade, and it was apprehended that this observation was rather less exact than the preceding ones.
There is every reason to suppose, that in the above experiment, the zenith distance of the signal was ob tained within a small fraction of a second.
To observe the angle between two signals, the plane of the instrument must first be brought into the plane of the two objects. To accomplish this, first set the tri pod of the instrument, with one of its feet as near as you can guess, in a line with that object, which of the two you judge to be nearest to the horizon; ancl, with the plane of the circle vertical, and the lower teloscope hori zontal, (both to the exactness of two or throe minutes') bring the telescope to the object, partly by tu•.iing in azimuth, and partly by screwing the foot screw ; next turn the circle round upon the cross axis of the stand, Vic of the a5ove Table.
With the distance of the object which is on the same side as the eccentric telescope, that is, the distance or the left hand object in our instruments, enter the Table and take a correction, to which you prefix the sign + ; with the distance of the right hand object, take a second cor rection, which is to have the sign minus —.
Example. Suppose thc distance of the left hand ob ject to be 5000 fathoms, and the right hand object to be distant 22,000 fathoms, and the eccentricity to the left ; The above correction arises from the mechanical construction of the instrument ; but the principal cor rection is to reduce the observed oblique angle to the horizontal angle, for which purpose thc subjoined Tables are added. They were calculated by the French as tronomers, who have endeavoured by every means in their power to extend and facilitate the use of this valua ble instrument.
Let A= the angle of position, or the observed angle. II = the altitude of the signal A.
h =the altitude of the signal B.
I.et 71= sin.' 3 (11+h) tang. 3 A — sin.' 3 (H—h) cot. 3 A.
Then thc cor. x = n, sec. 11, sec. h.
If the zenith distances differ more than 2° or 3' from 90°, the following formula may be employed.
C+ ;-I Sin. sm. o — 3 sin. 0"— being the angle reduced to the horizon, C the angle at the centre, ;and the zenith distances of the signals.
To facilitate this reduction, we have added the tables calculated for this purpose by M. Delambre. By these tables we may at the same time reduce the horizontal angle to that formed by the chords.
The use of these tables will be easily understood by an example.
h is the sum of the zenith distances of the ob served objects diminished by 180° If the sum should be less than 180°, II + h is thc re mainder required to complete 180°.
— h is always the difference between the two zenith distances.
(II + h) and (II — h, are always considered as positive numbers.
+ Q is the sum of the distances in French toiscs be tween the observer and each of the signals.
P — Q is the difference between these distances; (P — Q) is always positive.
With (I' + Q) and (1)— Q) take in Tab. II. two num bers, to which you always must annex the sign.
With the observed angle, take in Tab. IV. the num ber, Tangent, to which the sign -I- must be always an nexed, and which must be placed under the factor found by 1-1 +h.
With the same angle take in the adjoining column, Cotangent, to which annex sign—, and place it under the factor found by II— h. Place these same numbers under the factors (P + Q) and (I' —Q), as in the ex ample. Make the four requisite multiplications.
The difference of the two first products is the reduc tion to the horizon, to be applied according to its sign.
'clic difference of the two last products is the reduC tion to the chords, to be applied NVith its proper sign to the horizontal angle.
This last reduction is almost always subtractive, but it sometimes becomes additive, by the fourth product ex ceeding the third.
In genet al, the fourth product is nothing, and the third always very small ; so that in calculating the re duction, which is indispensable, it is very little more trouble to reduce the angles to the chords. These ta bles are, in general, quite sufficient for thc reduction to the horizon ; but, for greater exactness, 'Cable III. is added. The difference of the products, as obtained above, may, by means of this table, be multiplied by sec. II, sec. h, as required by the formula. If greater precision be required, the whole calculation may be repeated with the corrected angle, instead of the ob served angle.
To make use of this table, it is necessary to have a plan of the triangles with a scale. The arguments arc on one side of the triangle as a base, and the height.