240 PHILOSOPHICAL TRANSACTIONS. [ANNO 1797. 



of reducing a spheric angle to the angle between the chords, the spheric angle will 

 be represented by a, and the angle between the chords by a = a -f- da ; and da = 

 4-'*, \a . vs, (h - h) — 4-f, \a . vs, (h + h) — vs, da . 't, a (if v,d represent the arcs 

 to the chords) = 4^,4-a . vs, 4 (d ^ c?) — $t, 4^z . us, 4- (d -f d) — w, da . 't, a ; 

 A = a — (4^, 4-a . ttf, 4- (d + a 1 ) — 4-'/, 4-a . t'^4- (d ^ d) — w, aa . 't, a ; where the 

 last term will change its sign to affirmative, if a is greater than 90 . If the answer 

 is required in seconds, the correction must be multiplied by 206265, the number 

 of seconds in an arc = radius. The calculation will be easily made by logarithms. 



Practical rule. — The practical rule deduced from the above conclusions is the fol- 

 lowing, and given in the words of the Astronomer Royal. " To the constant lo- 

 garithm 5.0134 add l . t, \a and l . vs (d + d ;) the sum diminished by 20 in the 

 index is the logarithm of the first part of the value of da in seconds, which is always 

 negative. To the constant logarithm 5.0134 add l . t', 4a, and l . w, 4- (d ^ d), 

 the sum diminished by 20 in the index, is the logarithm of the 2d part in seconds, 

 which is always affirmative. These 2 joined together, according to their proper 

 signs, will give the approximate value of da. To its logarithmic versed sine, add 

 •L.t',a and constant logarithm 5.0134, the sum, diminished by 20 in the index, 

 will be the logarithm of the 3d part in seconds, which will be negative or affirma- 

 tive, according as a is less or more than 90 . This applied according to its sign, to 

 the approximate value of da, will give the correct value of da. If the 3d part comes 

 out considerable, it should be computed anew with the last value of da. The value 

 of da } finally corrected, applied to a, will give a, the angle between the chords." 



In the application of the above rule, to the computation of such corrections as 

 may be applied to the angles of any triangles in this survey, it is manifest that the 

 last step may be entirely neglected, on account of the smallness of the approximate 

 value of da, whose versed sine is one of the arguments. Being therefore confined 

 to the use of the first 2 steps, the operation is very short. An example is here 

 given in the computation of the correction for reducing the angle at Chancton- 

 bury Ring in the 39th triangle, given in the last account, to that formed by the 



chords. 



EXAMPLE. 



Constant logarithm 5.0134 5.0134 



Log. tang. \a = 78° 06' 10.7112 Log. co. tang. %a 9.2887 



Log. vs . \ (h + K) = 19/ 53".5 . . 5.2237 Log. M . \ (h - h) = 5' 53".5 4.1669 



0.9483 + .8".88 -2.4690 + ©".03 



1st correction — 8.88 

 2d correction + 0.03 



— 8.85 the correction required. 



After this follows a calculation of the sides of the great triangles, carried on from 



the termination of the series, published in the Philos. Trans, of 1795, along the 



coasts of Dorset, Devon, and Cornwall, to the Land's End ; not necessary to be 



here inserted. Then follows a series of observations of the angular elevations and 



