Longitudes of Dunkirk and Paris. 241 
meridian at Greenwich, and the angle EBG will be the lati- 
tude of Greenwich, or 51 0 28' 40". Let HO (parallel to 
GA) be the feftion of the parallel to that perpendicular, palling 
through Dunkirk. Then by the Table, p. 232. (Vol. LXXX.) 
the arc GH is 152549 feet; but this arc exceeds the real dis- 
tance of the parallels GA, HO, not more than a fathom ; 
therefore this diftance may be taken 25424 fathoms. Now 
the fedtions GA, HO, of the ellipfoid being fimilar, from the 
known properties of the figure, we fhall get HO the (hotter 
axis of the fe£tion of the parallel = 6959396, its longer axis 
— 6979374, and HW=353i757 fathoms, W being the point 
where HO cuts the axis PI of the ellipfoid. Hence, if D be 
Dunkirk, and the arc HD the meafured arc of the parallel, we 
have given the length of this arc, or 547058 feet (Table, p. 
232.) = 91176 fathoms, and alfo the point W in the leffer axis 
of the fe&ion HO, to determine the angle HWD in the plane 
of this fe&ion. But reverting the feries which exhibits the 
length of an elliptical arc in terms of th tabfeifs and ordinate , 
will be of little ufe in the prefent cafe, where the arc and its 
chord are very near of the fame length : For, let HKOL (fig. 
2.) be the feftion of the parallel, where HO = 6959396, and 
KL = 6979374, are the axes; and HW = 353i757, as in fig. 
1. ; alfo, fuppofe 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 deferibed with the radius HW, or with HS (3499700) 
and thence determine the angle SWD from the twofides SD, 
SW, and the included angle (the fupplement of HSD); in 
either cafe we get the angle HWD the fame, or i° 28' 44 // ,8 
to within 1 . This angle being obtained, the inclination of 
the planes PHW, PDW (the planes of the meridians of 
Vol, LXXXL K k Green- 
