70 PHILOSOVHICAL TRANSACTIONS. [aNNO 1791' 



= 6979374, and HW = 3531737 fathoms, w being the point where ho cuts 

 the axis pi of the ellipsoid. Hence if d be Dunkirk, and the arc hd the mea- 

 sured arc of the parallel, we have given the length of this arc, or 547058 feet, 

 = 91176 fathoms, and also the point w in the less axis of the section ho, to 

 determine the angle hwd in the plane of this section. But reverting the series 

 which exhibits the length of an elliptic arc in terms of the absciss and ordinate, 

 will be of little use in the present case, where the arc and its chord are very near 

 of the same length: For, let hkol (fig. 14,) be the section of the parallel, 

 where ho = 6959396, and kl = 6979374, are the axes; and hw = 3531757, 

 as in fig. 13; also, suppose hs is the radius of curvature at h, or at the middle 

 of hd; then, if we conceive the arc hd to be a right line, or described with the 

 radius hw, or with hs (3499700) and thence determine the angle swd from the 

 two sides sd, sw, and the included angle (the supplement of hsd;) in either 

 case we get the angle hwd the same, or 1° 28' 44". 8 to within l". This angle 

 being obtained, the inclination of the planes phw, pdw (the planes of the me- 

 ridians of Greenwich and Dunkirk, fig. 13,) or the longitude of the point d, 

 will be found by the common proportion which in a right-angled spherical tri- 

 angle determines an angle when the legs are given; this will be obvious by con- 

 ceiving a sphere, of any magnitude, to be described about w as a centre. 



Hence, as rad. : cotang. angle hwd (1°28'44".8) :: sine angle hwp (38° 

 31' 20") : cotang. 2° 22' 26"i, the inclination of the planes of the meridians ph, 

 PD, or longitude of Dunkirk on this ellipsoid. And as the difference of meri- 

 dians of Paris and Dunkirk is 2' 2l".9 (for this will not be materially affected by 

 different hypotheses) the longitude of Paris will be 9"' 204^ in time. The lon- 

 gitude of Dunkirk from Paris (2' 21". 9) is the mean longitude deduced in 

 vol. 80, which is only l".l less than that given in the Connoissance des Temps, 

 1788. 



The method of computing the latitude of the point d, were it necessary, is 

 thus: as rad. : cosine dwh :: cosine hwp : cosine dwp; and since the point w 

 in the axis pw is given, and also the angle dwp in the plane of the meridian pd, 

 by the foregoing proportion, the point d will be determined by the properties of 

 the ellipse; which in fact is nothing more than finding the inclination of the ver- 

 tical at the point d with the given line dw, which inclination added to the angle 

 DWP, gives the co-latitude of the point d. And hence may be evinced the truth, 

 that if the value of an arc on a spheroid, considered as an arc of a great circle 

 perpendicular to the meridian, be given, the longitude may be found by spheriail 

 computation, but not the latitude. For conceive the arc hd to be perpendicular 

 to the meridian at h, then the angle hwp would be the co-latitude of the point h; 

 and the former proportion would give the longitude of d, whether the figure was 

 a sphere or spheroid ; and the angle dwp, found by the latter proportion, would 



