1^ ON THE FIGURE OF THE EARTH. 



Solution of the J) 



problem for fin. f d! = - — : find the angle w' contained between the polar 



finding the 2a ° *^ 



figure of the diftances, and the difference of longitude wwill be — u'4- 



earth from the ,_ r i ja /r ^» • /• a»i /• -. 



length of a chord if ^ cof. f d^ (fin. X^ -f fin. <?'^)-2 fin. X fin ^ , 



joining two a fin. w' 



^lowu p aces, ^^^^ latitude <P may alfo be found from the fame equation, 

 when X, u and D are given. For if the bafe of a fpherical 

 triangle be zz d, the two other fides — 90° — X, 90° — (p^ and 

 the contained angle ~ w, the cof. u will be ZZ 



D* 



1 ~^— 5 — fin. X fin. <Pf 



1^^.^^^^^^, Now let <^zz((>'-\-x, where x mufi: 



be very fmall, and cof. X cof. <?>' there refults cof. w' zr cof. w -f- 

 (cof. X fin. <?' cof. w — fin. x cof. (p') 



— ;: ;:: — ; X fin. x: confequently 



col. X cof. (f)' n. J 



(cof. A fin. <p' cof. w — fin. X cof. ffi') ^ c 



— ; X fin. ar — — 



col. X col. <?•' a 



( 2 (fin. X — fin. (py (fin. X* + fin. (^'^) \ nearly, 



and X = - X 

 a 



eof. Xcof. <p' (2 (fin. X-fin. (?')' (^^n- ^* + ^n- 'P'^)') 



cof. X fin. cp' cof. a — fin, X cof. <?' 

 From the inveftigation of Mr. Playfair's problem, therefore, 

 we have obtained very accurate rules for finding w from X, ^ 

 and D, and (pixom x, w, D. 



Now in order to find an equation exprefling the relation be- 

 tween the latitudes, difference of longitude, and one of the 

 azimuths, let AL be perpendicular to the meridian PAO in 

 A meeting FE in L, and BK perpendicular to AL. Join KE, 

 and the angle BKE will be equal to the fpheroidal angle OAB, 

 and BFE equal to the angle APB or difference of longitude. 

 Let OABzz BKE — A, BFE ZZ w, and x, (pas before, then will 

 LEbe — (CD — CF) cotang. x -j- FE — AD,KE::z (CD — 



KF 

 CF) cof. X -f (FE — AD) fin. x, and cotang. A ~~~' 



BE 



Whence by fubfiitutlng the valves of CD, CF, AD, BF given 



above, and rejeding the powers of c higher than the firfi, there 



refults 



