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towards the observer at a particular time, the most essential service was derived from its use; and the consequence was, the invention of a more perfect instrument, of which a description is given, accompanied with a drawing.

The second method consists in the exhibition of a very brilliant light at night. At the commencement of the Survey of England, General Roy had recourse, on several occasions, and especially in carrying his triangles across the Channel, to the use of Bengal and white lights; for these, parabolic reflectors illuminated by Argand lamps were afterwards substituted as more convenient; but from want of power they appear in turn to have gradually fallen into disuse. With a view to remedy this detect, a series of experiments was undertaken by Lieutenant Drummond, the result of which was the production of a very intense light, varying between 60 and 90 times that of the brightest part of the flame of an Argand lamp.

This brilliant light is obtained from a small ball of lime about 3-8ths of an inch diameter, placed in the focus of the reflector, and exposed to a very intense heat by means of a simple apparatus, of which a description is given, in his account. A jet of oxygen gas directed through the flame of alcohol is employd as the source of heat. Zirconia, mag. nesia, and oxide of zinc were also tried; but the light emanating from them was much inferior to that from lime. Besides being easily procured, the lime admits of being turned in the lathe, so that any number of the small focal balls may be readily obtained, uniform in size, and perfect in figure. The chemical agency of this light is remarkable, causing the combination of chlorine and hydrogen, and blackening chloride of silver. Its application to the very important purpose of illuminating light-houses is suggested, especially in those situations where the lights are the first that are made by vessels arriving from distant voyages.

Both the methods now described, for accelerating geodesic operations, were resorted to with much success during the season of 1825 in Ireland; and on one occasion, where every attempt to discern a disant station had failed, the observations were effected by their means, the heliostat being seen during the day, when the outline of the hill ceased to be visible, and the light at night being seen with the naked eye, and appearing much brighter and larger at the distance of 66 miles, than a parabolic reflector, of equal size, illuminated by an Argand lamp, and placed nearly in the same direction, us an object of reference, at the distance of 15 miles.

15. In proof of the great correctness of the English sur

vey, we shall state a very few particulars, besides what is already mentioned in art. 12.

General Roy, who first measured the base on Hounslow. Heath, measured another on the flat ground of Romney. Marsh in Kent, near the southern extremity of the first series of triangles, and at the distance of more than 60 miles from the first base. The length of this base of verification, as actually measured, compared with that resulting from the computation through the whole series of triangles, differed only by 28 inches.

General Mudge measured another base of verification on Salisbury-Plain. Its length was 36574-4 feet, or more than 7 miles; the measurement did not differ more than one inch from the computation carried through the series of triangles from Hounslow-Heath to Salisbury-Plain. A most remarkable proof of the accuracy with which all the angles, as well as the two bases, were measured!

The distance between Beachy.Head in Sussex, and Dun. nose in the Isle of Wight, as deduced from a mean of four series of triangles, is 339397 feet, or more than 64 miles. The extremes of the four determinations do not differ more than 7 feet, which is less than 1 inches in a mile. Instances of this kind frequently occur in the English survey *. But we have not room to specify more. We must now proceed to discuss the most important problems connected with this subject; and refer those who are desirous to consider it more minutely to General Mudge and Colonel Colby's "Account of the Trigonometrical Survey ;" Mechain and Delambre, "Base du Systême Métrique Décimal;" Swanberg, “Exposition des Opérations faites en Lapponie ;" and Puissant's works entitled "Geodesic," and "Traite de Topographie, d'Arpentage, &c."

* Puissant, in his “Geodésie,” after quoting some of them, says, "Neanmoins, jusqu'à présent, rien n'égale en exactitude les op rations géodesiques qui ont servi de fondement à notre système métrique." He, however, gives no instances. We have no wish to depreciate the la bours of the French measures; but we cannot yield them the preference oa mere assertion.

SECTION II.

Problems connected with the detail of Operations in Extensive Trigonometrical Surveys.

PROBLEM I.

It is required to determine the most advantageous
conditions of triangles.

1. In any rectilinear triangle ABC, it is, from the propor tionality of sides to the sines of their opposite angles, AB : sin c sin A, and consequently Aь. sin

BC

A BC. sin c.

Let AB be the base, which

is supposed to be measured without percep

C

tible error, and which therefore is assumed

as constant; then finding the extremely A

B

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small variation or fluxion of the equation on this hypothesis, it is AB. COS A. A sin c. BC + BC. cos c. c. Here, since we are ignorant of the magnitude of the errors or variations expressed by a and c, suppose them to be equal (a probable supposition, as they are both taken by the same instrument), and each denoted by v: then will

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or finally, BC = v. BC (cot A - cot c).

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This equation (in the use of which it must be recollected that v taken in seconds should be divided by R", that is, by the length of the radius expressed in seconds) gives the error BC in the estimation of BC occasioned by the errors in the angles A and c. Hence, that these errors, supposing them to be equal, may have no influence on the determination of BC, we must have a = c, for in that case the second member of the equation will vanish.

2. But, as the two errors, denoted by A, and c which we have supposed to be of the same kind, or in the same direction, may be committed in different directions, when the equation will be вc±v BC (cot Acot c); we must inquire what magnitude the angles a and c ought to have,

so that the sum of their cotangents shall have the least value possible; for in this state it is manifest that BC will have its least value. But, by the formulæ in chap. 3, we have

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And hence, whatever be the magnitude of the angle в, the error in the value of BC will be the least when cos (AC) is the greatest possible, which is, when a = c.

We may therefore infer, for a general rule, that the most advantageous state of a triangle, when we would determine one side only, is when the base is equal to the side sought.

3. Since, by this rule, the base should be equal to the side sought, it is evident that when we would determine two sides, the most advantageous condition of a triangle is that it be equilateral.

4. It rarely happens, however, that a base can be commodiously measured which is as long as the sides sought. Supposing, therefore, that the length of the base is limited, but that its direction at least may be chosen at pleasure, we proceed to inquire what that direction should be, in the case where one only of the other two sides of the triangle is to be determined.

Let it be imagined, as before, that AB is the base of the triangle ABC, and BC the side required. It is proposed to find the least value of cot a cot c, when we cannot have a = c. Now, in the case where the negative sign obtains, we have

AB-BC. COS B
BC. sin B

BC-AB COS B
AB. Sin B

=

A B2 - BC3 AB. BC. Sia B

cot A- cot c = This equation again manifestly indicates the equality of AB and BC, in circumstances where it is possible: but if AB and BC are constant, it is evident, from the form of the denominator of the last fraction, that the fraction itself will be the least, or cot a - cot c the least, when sin в is a maximum, that is, when в = 90°.

-

5. When the positive sign obtains, we have cot a+cot c= cot A +

(RO2-AR2 Sin2 )

AB SI A

cot a +(

EC2

1).

AB SIU A

Here, the least value of the expression under the radical sign, 90°. And in that case the first term, Therefore the least value of cota + 90°; conformably to the rule given de la Terre, p. 88). But we have

is obviously when a cot A, would disappear. cot c, obtains when a by M. Bouguer (Fig, VOL. II.

11

already seen that in the case of cot A-cot c, we must have
B90 whence we conclude, since the conditions A =
= 90o,
B = 90°, cannot obtain simultaneously, that a medium result
would give a = B.

If we apply to the side Ac the same reasoning as to BC, similar results will be obtained therefore in general, when the base cannot be equal to one or to both the sides required, the most advantageous condition of the triangle is, that the base be the longest possible, and that the two angles at the base be equal. These equal angles, however, should never, if possible, be less than 23 degrees.

PROBLEM II.

To deduce, from angles measured out of one of the stations, but near it, the true angles at the station.

A

When the centre of the instrument cannot be placed in the vertical line occupied by the axis of a signal, the angles observed must undergo a reduction, according to circumstances. 1. Let c be the centre of the station, B P the place of the centre of the instru ment, or the summit of the observed angle APB it is required to find c, the measure of ACB, supposing there to be known APB ⇒ P, BPC = P, cr =d,

BCL, AC = R.

P

Since the exterior angle of a triangle is equal to the sum of the two interior opposite angles (th. 16 Geom.) we have, with respect to the triangle IAP, AIB = P + IAP; and with regard to the triangle BIC, AIB = c + CBP. Making these two values of AIB equal, and transposing IAP, there results C = P + IAP — СВР.

But the triangles cap, CBP, give

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d. sin (p+p)

;

R

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And, as the angles CAP, CBP, are, by the hypothesis of the problem, always very small, their sines may be substituted for their arcs or measures: therefore

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