VOL. LXXXr,] PHILOSOPHICAL TRANSACTIONS. 71 



be the co-latitude of d supposing it a sphere, in which case the point w becomes 

 the centre; but this will not hold in a spheroid, because dw would not be per- 

 pendicular to the meridian at d. 



The foregoing method of computing the longitude from the measured arc of 

 a parallel on a given ellipsoid, though evidently the direct one, will be tedious, 

 especially when the lengths of the measured arcs, gh, hd, are very considerable. 

 But when the latitude of the point h is determined from the measured arc gh, 

 on the known meridian, and the extent of the other arc hd, or rather the angle 

 HWD, is not more than 2 or 3°, the same conclusions, extremely near, may be 

 obtained in the following manner, which is nearly the same as the method used 

 in computing the longitudes in the table of general results, vol. 80. 



Suppose G and d (fig. 13,) to be Greenwich and Dunkirk; ph, vd, their me- 

 ridians, as before; and let hd, instead of its being a parallel to the perpendicular 

 at Greenwich, be an arc of an ellipse cutting the meridian of Greenwich at right 

 angles, suppose in the point h. Then the arc gh being = 152549 + 30 feet 

 nearly (because the ellipse which passes through d, and is at right angles to the 

 meridian pg, will fall about 50 feet to the south of the point cut by the parallel,) 

 therefore the value of the arc gh, or 25433 fathoms, will, on this ellipsoid, be 

 25'4".4, and consequently the angle vwh, or the co-latitude of h, is 38° 56' 

 24'''.4. Now, the radius of curvature of this perpendicular ellipse at h, the ex- 

 tremity of the lesser axis, will be 3499798 fathoms*, which divided by 57.2Q577Q, 

 &c. the degrees in the circular arc which is equal to the radius, gives 6 1083 

 fathoms for a degree on this ellipse, considered as a great circle perpendicular to 

 the meridian at the point h on the ellipsoid; and since the length of this arc 

 hd will be nearly the same as that of the parallel, or 9117O fathoms, its value 

 will be 1°29'33".6, the arc dh, or rather the angle dwh. Hence, as rad. : 

 cotang. 1°29'33".6 (hwd) :: sine 38° 56' 24^.4 (hwp) : cotang. 2°22'26".8, the 

 longitude of d, or Dunkirk, the same as before, very near; hence the longitude 

 of Paris will be 2° 20' 4".9. But the same may be obtained from the mean dis- 

 tance of the meridians of Greenwich and Paris, or 537950 feet. 



It appears from the foregoing hypothesis, that the measured degrees of the me- 

 ridian in middle latitudes will answer nearly on an ellipsoid whose axes are in the 

 ratio assigned by Sir Isaac Newton. But this will receive further confirmation 

 from the 5th ellipsoid, vol. 80, where the near agreement between the computed 

 and measured arc of the meridian between Greenwich and Perpignan (differing 



* It is not necessary to determine the axes of this ellipse, because when hw is perpendicular to the 

 curve of the meridian, it will, by the nature of the figure, be tlie radius of curvature of the arc hd 

 at the point 11. Hence, if we put v for the cotang. and c for the cosine of tlie latitude of the point 



H, and let a denote the sine of an arc whose tang, is — x r; then - x ce = hw, by the proper- 

 ties of the ellipse. 



