and 



152 PROBLEMS IN SPHEROIDAL TRIANGLES. 



cof. A col. <? fill ci' 2 <y^ 



. 1. — - r ^r — 7—^= 4- tang. X tang, (px ,. \ = 



col. X cof. (?) fin u)' b o ^^^^' 



2 tang. X tang. <? fin ^dx—r; wherefore Q is equal to 4 Cm 



-^X • — ; 2 tang. X tang, (y fin | dx -r* a"^ 



2 w * w' 



w=:w'-3. ■< tang. X tang, ^x -7 ■^ — 7--- ^ nearly ; where 



the corce6lion Q^ will be given in feconds of a degree, if d, 



u, and (K—<P), be expreflfed in feconds. 



„ ^ , . r , cof. rf— fin X fin <? , ^ , 



By fpherics cof. w'= , whence we hnd, 



■ ■' ^ cof. X cof. (p 



when d and w' are fmall, u'zi ^ { — ^ r-i i 



V L col. X col. <?» J 



without any fenfible error, which value fubftituted in ihe for- 

 mula w=;w'-f-Q J, will give the true difference of longitude re- 

 quired. 



When the meafured chord is perpendicular to the meridian, 



(p is nearly equal to X, and confequently w'= — ;: — = — , 



"^ col. X a col. X 



andw= J— ■x(l-3finX»), or c^= ; but 



a col. X col.X (a-fcfinX^) 



c-f-c fin X* is equal to the radius of curvature of the perpendi- 

 cular to the meridian, in the latitude X=AM in the figure 

 (Journal for May), Which put = R, and there refiilts « ~ 



|r j: — , which correfponds exa6^1y with one of Legendre's 



theorems^ (Mem. Acad. 1787). 



Example. Let X=50<> 44' 23",71, the latitude of Beachy 

 Head, <?r:50° 37' 7",31, the latitude of Dunnofe, D = 

 56566,57 fathoms, and <i=3496740. Then becaufefin | d~ 



--, d'\%z=— neaTly==3^36'',73, o>'=\^ 27'0",65, and «=«' 



-3105",74 5; in Trhich^Hf-we fuppofe 5= , wefhalihave 



ffzzl* 26'47",93, 



a. * Having 



