CIRCLE. 



503 



Circle. 



Use of ike above Table. 



With the distance of the 'object which is on the same 

 side as the eccentric telescope, that is, the distance of 

 the left hand object in our instruments, enter the Tab'e 

 and take a correction, to which you prefix the sign + ; 

 with the distance of the right hand object, take a second 

 correction, which is to have the sign minus . 



Example. Suppose the 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 ; 



Fathoms. 



Dist. of object to the left 5000 

 right 22,000 



+ 0".40 

 0"09 



Total corrrection + 0".31 



The above correction arises from the mechanical con- 

 struction of the instrument ; but the principal correction 

 is to reduce the observed oblique angle to the horizontal 

 angle, for which purpose the subjoined Tables are added. 

 They were calculated by the French astronomers, who 

 have endeavoured by every means in their power to ex- 

 tend and facilitate the use of this valuable instrument. 



Let A = the angle of position, or the observed angle. 

 H= the altitude of the signal A. 

 A = the altitude of the signal B. 



Let it = sin. 1 .J (H + /<) tang. ! A sin. 1 \ (H A) 

 cot. A. 



Then the cor. x =: n, sec. H, sec. h. 



If the zenith distances dinx-r more than 2 or 3 from 

 90, the following formula may be employed. 



. /C+3+3' A . /C+3 + 3' v \ 



. . / mn.( 7T- 3) sin. I ! 3') 



Sin.z = / V 2 / V 2 t_ 



sin. 3 sin. 3' 



2 being the angl.- reduced to the horizon, C the angle 

 at the centre, 3 and & the zenith distances of the signals. 



To facilitate this reduction, we have added the tables 

 calculated for this purpose by M. Dclambre. 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 + h is the sum of the zenith distances of the ob- 

 served objects diminished by 180. 



If the sum should be less than 180, H + h is the re- 

 mainder required to complete 180. 



H h is always the difference between the two ze- 

 nith dis^ncJs. 



(H + A) and (H A) are always considered as po- 

 sitive numbers. 



P + Q is the sum of the distances in French toises 

 between the observer and each of the signals. 



P Q is the difference between these distances ; 

 IP Q) is always positive. 



With (P 4 QJ and (P Q) take in Tab. II. two 

 number! 1 , to which you always must annex the sign. 



W"h the observed angle, take in Tab. IV. the num- 

 ber, Tangent, to which the s:gn -f- must be always an- 

 nexed, Hud which must be placed under the factor found 

 by H + h. 



With the same angle take in the adjoining column, 

 Cotangent, t-> which annex sign , and place it under 

 the factor found by H A, Place these same numbers 

 under the factors (P -f- Q) and (P Q), a> in the ex- 

 ample. Mak<- the four requisite multiplications. 



The diffen r.oe of the two first products is the reduc 

 tion to the horizon, to be applied according to its sign. 

 1 8 



The difference of the two last products is the reduc- Circle, 

 tion to the chords, to be applied with its proper sign to * T^ 

 the horizontal angle. 



This last reduction is almost always ubtractive, but 

 it sometimes becomes additive, by the fourth product ex- 

 ceeding the third. 



In general, 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 the reduction to-' 

 the horizon ; but, for greater exactness, Table III. is. 

 added. The difference of the products, as obtained 

 above, may, by means of this table, be multiplied by sec. 

 H, sec. A, as required by the formula. If greater pre- 

 cision be required, the whole cakulation may be repeated 

 with the corrected angle, instead of the observed angle. 



To make use of this table, it is necessary to have a 

 plan of the triangles with a scale. The arguments are 

 on one side of the triangle as a base, and the height. 



Observed Angle. 

 32 2(y 15".7 



Example. 



Zenith Distance. 



A = 89 41' 54".6 

 B = 88 49 15.6 



178 31 10.2 



Distance of Signal i. 

 in Toises. 

 A= 18283 

 B = 24423. 



H + A= 1 28 49.8 

 H A= 52 39 



Tab. I. 



Tab. II. 



Argument H + A H A 



Factors +1.669 +0.587 



Tab. IV. + 5.99 7.106 



15021 35'22 



15021 0587 



8345 4109 



+ 9.99731 41.71222 



Observed angle . . . 

 Reduction to horizon . . . 



Horizontal angle ....=: 

 Reduction to the chords . 



0.112 0.003 

 + 5.99 7.106 



0.67 +0.2 



+ 0.02 .- 



0.65=.reduction 



to the chords. 



32 20' 15".7 

 31.7 



32 19 44.0 

 0.65 



Angle of the chords .... 32 19 43.45 



When the depressions are small, 

 the tables, use thix formula. 



we may, instead of 



sin. 1", 



Example by the Formula. 



3 = 89 41' 54".6 

 y= 88 49 15.6 



3 + 3' =178 31 10.2 3 3' = 52' 39" 

 90 -~= 44 24.9 ^-=0 2619.5 



s= 2664".9 = p = Z579".5=q 



