FROBLEMS IN SPHEROIDAL TRIANGLES. 153 



2. * Having given the latitude of a place, and its longitude Prob.II. Givca 



and diftance from another place, it is required to find the other j"^^J*^'^^"j.j.^ 



latitude ?' tance from an- 



Let X be the given and ? the required latitude ; D the mea» °^*'" P'^==- , . 

 o ^ ' Req. lat. of this 



fured dtftance, d=: — , « the difference of longitude, and <P' ^ ' 



the latitude on the fphere found from \ «, d. Then by fphe- 

 rics cof. A cof. <^' cof. w-f-fin ^ fin <?^=;cof. d, whence, when 

 d, 4>, and X— ?>'are fmall, there refalts X — <?»'=3: -/(rf*— cof. A 

 cpf.(p'. t.,»)=±iy/(<i* — w* cof. X*) nearly. But by the theorem, 

 page 16, if ^=(p'-\-x, we find j:=Q^, in which the variable 



quantity Q is =-^ ; — - — ? ^^ ~ — =» 



•' col. X fin <?' col. ty—im X col. (?>' 



f d« (finA^+fin r')-2cofY^4-^'y (X-*?')* 



"^ ^ cof. A fill ^' tin fc'^-j-lin (X-!?') ' 



f rf* (finX*+fin^'*)-2cof. /^li^V (A-(p')« 



0= ^ ^ ^ . 



- cof.Afin?'fini«'x«'+(X-(p') 



Now when X and ?are nearly equal, this formula becomes, 

 _ fin X«. rf' - 2 cof. A^ (X - (pQ* _ 

 ~~ w. cof. \ fin <?' liu'i«4-(x — <?') "" 

 fin X^ <j* - 2 cof. x" (X - <P^)* 

 cof. X fin X fin fw. u-\-{\—<p') 



3. 'Having given the latitude X, the horizontal angle a, Prob. in. 



and the diflance D, it is required to find the difference of Ion- ^'^^" o"^ 'at, 

 -^ J ., with the horiz. 



g'tude ?' angle ^nd dif- 



Let <P be the latitude, aqd i* the longitude of the place re- "nee; To find 



jy diff. long. 



quired; with the latitudes x, ?>, and diftance J= — , on the 

 ♦ a 



fphere, to find the difl^erence of longitude «', and the horizontal 



angle a' at x. 



«,. ... /- , fin rf fin a' fina'. c£ , , „ ^ 



Then will fin « = r-r—, or w = ■ ' ; but by firft 



. > , , col. <P Chi. <p ^ . 



, A . d* 2rx-(p)«n 



queftien w=w — S j tang, x tang. ? X -7 ^jT— | ; 



d fin a' C -/(ang. X fin ©cof. (f> 

 wherefore w=— -7 — . ■? 1— 3( . Vi ' . ■ "'v ^ "" 



col. <P (^ \ 111) «-'*^,-^ > 



2cof. (p*(x-(;))* \ -j _ rffin«- ^ gX finX* 



rf^fina'^ / i ~ col. (? i. ^ ^^lin*'*'~ 

 2cof. X» (X-(p)' 



■^)l 



4»fin«'» y5' l^ow. 



