VOL. XC.] PHILOSOPHICAL TRANSACTIONS. 7g3 



81° 56' 53", from observation, which may be considered the same with that found by this mode of com- 

 putation. In all cases in which the data were equally correct, no doubt the direction of meridians might 

 be computed, without fear of the results deviating much from the truth ; but if it be required to find 

 the angle at Black Down, from the observed direction of the meridian at Dunnose, a different method 

 must be used. It is however less accurate than the former one, and it has been expressly for this reason, 

 that I have not introduced this subject into the account. 



In the adjoining diagram, suppose b to be Black Down; D, Dunnose, and 

 n, Nine Barrow Down; also let pb, the meridian of Black Down, be pro- 

 longed to m, and dm be drawn, fm being = pd. Then we shall have 3 

 spherical triangles, bpd, bnd, bmd. Now the angle nbd was found from 

 observations to be 4° 30' 28", and bnd 172° 27' 33". 5; these give the angle 

 bdn = 3° 1' 59". 5, nearly, because the excess of the 3 angles above 180° 

 is 1". The observed angle at d, Dunnose, between Nine Barrow Down 

 and the meridian dp, or pdn, was 87° 56' 53"; therefore 87° 56' 53" — 

 3° 1' 59".5 = 84° 54' 53".5, is the angle at d, between the meridian and 

 N * the station on Black Down. 



Now, the difference of longitude between b and n, or the angle at p, has been already found = 1* 

 20' 46".4; and since bp is very nearly = pd, and pd is small, we shall have rad. : tang. |r : cosine 

 dp : cosine bmd = 89° 28' 47". But the angle pdb has been found = 84° 54' 53". 5; therefore 89° 

 28' 47" - 84° 54' 53".5 = 4° 33' 53".5, the angle bdm; hence 180° </ 2" — 94° 2' 40".5 = 85° 57' 

 21". 5, or mbd; therefore 94° 2' 38",5, or dbp, is the angle at Black Down obtained in this way, 

 which differs nearly l6"from the observed one, viz. 94° 2' 22".75. It is probable, some portion of this 

 arises from defects in the observation made at Dunnose, on the lights fired at Nine Barrow Down; only 

 2 lights were seen; and as the observations differed 5" from each other, some degree of doubt exists, as 

 to the accuracy of the angle. The angle at Nine Barrow Down, between Black Down and Dunnose, 

 is not absolutely to be depended on for purposes of this kind, though there can be no doubt of its being 

 sufficiently near the truth, for that to which it has been before applied. In the correction of the angles 

 at that station, in our former accounts, we proceeded on the supposition of their being less satisfactory 

 than the other angles of the triangles to which Nine Barrow Down is a common station. For these rea- 

 sons, I am of opinion the computed angle cannot be applied as a test to the observed one; and it also 

 appears to me, that greater objections lie against similar comparisons between the computed and observed 

 angles at Butterton and St. Agnes ; as those stations could not be seen from each other, nor the latter 

 from Black Down. Though the computed directions of the meridians differ some seconds from the ob- 

 served ones, I am by no means doubtful of the truth of the latter; as the double azimuths of the pole 

 star, found from computation, agree very satisfactorily with those which have been used in obtaining the 

 directions of the several meridians. — In finding the value of the oblique arc, or the line which joins 

 Black Down and Dunnose, as used in the first method of computation, I have had recourse to the fol- 

 low? 



lowing correct expression, viz. d = ; — -: where dis the length of the required degree, p 



p + (wi — p) s* or 



that of the great circle perpendicular to the meridian, m that of a degree of the meridian itself, and * 

 the sine of the angle constituted by the oblique arc and the meridian. 



We next find a determination of the altitudes of the stations above the level of 

 the sea ; and the mean refractions deduced from observed angles of elevation and 

 depression. After a number of individual calculations, are then inserted the fol- 

 lowing tables of altitudes and refractions at several stations. 



Heights of the Stations, 



Stations. G ™"er a !T;k! 0VV ' Stations. Gro w 7* r a ™'k ,ow " 



Trevose Head Feet 274 Bodmin Down Feet 6'45 



St. Agnes Beacon 621 Black Down 1 ifto 



Hensbarrow 1034 St. Stephen's Down 605 



VOL. XVIII. 5 I 



