under one point of view for future use, we shall have the fol In calculating the distance between the parallels of latitude of two places, connected by means of a trigonometrical operation, regard must be had to their difference in longitude. If the triangles run nearly north and south, in which case stations must lie both east and west of the two meridians, it is sufficiently correct to proceed on the supposition of the earth's surface being a plane; but if, on the contrary, the triangles wholly diverge from the two meridians, or even partly do so, first running off obliquely and then returning again, a different of the degree oblique to the meridian; or, putting 1-s for c, and r for p-m, it will be Corol. If d be the length of the oblique degree, then, since d= ри we have s'd m and m = cdp And, if D be put for the length of another oblique degree at the same point, and S and C the sine and cosine of its inclination to the Sc⭑- C⭑s° x Dd, and p= S°c•— C's® meridian, we shall get m = × Dd, the meridional and perpendicular degrees, exhibited in terms of the oblique degrees combined with the sines and cosines of their inclinations to the meridian. Therefore, an ellipsoid may be determined from the lengths of two oblique degrees in the same latitude. We may likewise remark, from the nature of radii of curvature, at the same point G, that the expression will also give the oblique degree on different spheroids. pm method must be pursued. The necessity giving rise to this, originates from the radii of curvature of the oblique degrees continually varying, and the angles of convergency, between the several sides and their respective meridians, remaining unknown. It must be remembered, that the sides of the several triangles projected over the country, in this Survey, are not to be considered as so many distances on the earth's surface, but the lengths of the chord lines subtended by arcs. Therefore, it is manifest that, strictly speaking, all the chord angles should be used, and not the horizontal ones; with which, after the bearing of the first side with the meridian has been reduced to some plane beneath the earth's surface, a number of chord lines in the plane of that meridian are to be computed; the sum of which, augmented by the differences between those chords and their respective arcs, will give the true meridional distance. I have been at the trouble to calculate the distance between Clifton and Dunnose on this principle; and find the length of my arc to be 1036339,5 feet; which is, about 24 feet more than the distance determined by the other mode of computation. An advantage, however, attending a calculation on the principle now spoken of, is the ability of calculating, pretty nearly, the azimuth of any one station from an extremity of the arc. This, if the instrument with which the direction of the meridian is observed be not well divided, or otherwise not exactly fit for the operation, is necessary, and should be always done. The angle at Clifton, between Gringley on the Hill and the meridian, was observed to be 76° 17′ 25". According to my computation in the way spoken of, that angle is 76 17′ 30". A difference of 5", working all the way up from Dunnose through an arc of 2° 50′, is as small as can be expected, and serves to prove that the angles of the triangles, as well as the observed direction of the meridians, are consistent. I have given the meridional distance between Clifton and Dunnose, bearings of the sides, &c. deduced from the most simple of the two methods; first, because the result is sufficiently accurate; secondly, because it places within general reach, the means of examining this part of my operation. In attending to this remark, it must be remembered, that a line from Dunnose perpendicular to the meridian of Clifton, is only 4853 feet. SECTION SECOND. Operations at the Station on Dunnose, the Southern Extremity of the Arc, with the Zenith Sector. May and June, 1802. On the 8th of May, the circular or large theodolite was placed over the point selected for a new station: its distance was 61 feet from the gun, and in a direction due south. The following objects were then observed, the readings of which, on the graduated limb, were as follows. Sir R. WORSLEY'S obelisk (the top) LUTTRELL'S Folly Vane on the top of Portsmouth Church Sir R. WORSLEY'S obelisk, a second time 113° 14′28′′ 1 46 36,5 177 56 25 40 6 44,5 113 14 24,25 The above objects were observed, in order that no possible mistake might result; as (though not probable) accidental circumstances might have given rise to a wrong statement of the bearing of some one of the number, (except Portsmouth Church,) in the account of 1795. Omitting the obelisk, the bearings of the other objects, as extracted from that Paper, will If, from the readings on the limb, the angles between the obelisk and the other objects be taken, and applied to the lastmentioned bearings, we shall get the angle between the obelisk and the meridian, 87° 42′ 40′′ 35 45 Mean, 87° 42′ 40′′. May 9th. Erected the observatory, drove four long stakes into the ground, and brought their several heads into the same horizontal plane. Then erected the stand, set up the sector, and adjusted the axis level, and the axis itself; determined the exact weight the plumb-line would bear, and then examined how much the cross wires were out of their proper positions, as follows. The stand being firmly screwed down to the stakes, the sector was turned on its axis, till the pointed top of Sir RICHARD WORSLEY'S obelisk appeared in the field; it was then clamped to the azimuth circle, but subject to a small motion by turning an adjusting-screw. The pointed apex was then made to appear as just vanishing under the wires; in which situation of things, the side telescope was turned round, and laid in its several positions on the brass frame attached for its reception to the side of the sectorial tube; the top of the obelisk appearing as a vanishing point under the wires. On whichever face of its squares it was made to rest, the vernier of the azimuth circle read off to 84° 5′. The little telescope was then taken out of its frame, and the sector turned half round. It was then again introduced into its supports, and the interior stand moved, till the wires in the focus of the lateral telescope appeared on the obelisk as before. The vernier was then examined, which again stood at 84° 5'. This being settled, the sector was turned round, till its vernier stood at 176° 22' on the azimuth circle, in which situation, the plane of the divided arc was necessarily parallel to that of the meridian. The task of observation then commenced, and the performance of it was as follows. Observations made at Dunnose, to determine the Zenith Distance of B Draconis. |