ON THE FIGURE OF THE EARTH. 15 



From this equation the following method of determining (he Salution of the 



/igure of tl>e earth Is deduced. Let /be the length of a niea- ^^^^\^^i\^q^ 



iiired chord, and x, <p, u the latitudes and difference of longi- figure of the 



tude of its extremities j find S as above, and let ?« = 2 tin. l^, f ^'th from the 

 ' ' ^ length of a chord 



J r I Q ,-/• -va I r /*'»^ (^"- X — ^""'- '?)'• joining two 



and n = fin. f 3 (fin. A* + fin. r-) - '- j^^p^^ Inown places, 



' '^ &c. 



Then if we rejed all the powers of c higher than the firft, we 



Ihall have the fimple equation ma-^nc :=. I. Jn like manner 



find a fimilar equation m'a -j- w.''c = /', correfponding to any 



other chord whofe length is I', and there will relult a = 



n'l — nV , m'l—mV _, . . , 



» and c = ; ■ — . The approximation may be 



mn — m n mn —mn 



eafily carried futfher by including the fecond power of c, and 

 thus finding an equation of the form ma -}- nc -j- /?c *zzl; but 

 this labour would be ufelefs, as the method itfelf does not ad- 

 mit of greater accuracy. If (p = \ the equation becomes 



~r r-i "iza + cfin. (2)*, as is found by Mr. Playfair in 5 31. 



2 fill. 45 



From the firft equation a rule may be eafily derived for cal- 

 culating the difference of longitude of two places, when their 

 latitudes and diftance are given. For by tranfpofition we hav« 



"' (I -f- —(fin. X2 4- fin. $') n D-i -}- 4. flc (fin x-fin ^)% 



and by divifion, and rejeding the powers of c higher than the 



firft S^zrD^ -f-^(4a*(finX-fin.(p)*-D'(fin.x2-ffin.(i5*jV 



but 5* :«zr'2G2 (1— fin. xfin. ffl— cof. 7, cof. 1? cof. u), there- 



1— -— J— nn. A fin. up 

 fore cof.. =- '-';„,., ^„,^ -^(2(fin.^-f,n.<fr- 



— 2 (fi"- ^* +/!!''• ^^) ] and putting 1 ^ - fin. X fin.i?> 



col. A col. <P 



2(fin. A -fin. ^l"^ 



C (I 



= cof. u v.'e have ZZ w' -j x • — ■ -i^- — > 



a fin. w 



(finAM-Tin. (?*) 



■ , which ruje may be thus expreffed. Let 



there be a fpherical triangle, having two of its fides equal to 

 tlie polar diftancesof the places, and the third fide d fuch that 



fin. 



