rROBLENtS IN Sl>HEROIDA't TttlANOLEG. |^J[ 



iir. 



I'rohlems in Spheroidal Triangles. ^j^Peregrinus Proteus, 

 To Mr. NICHOLSON. 



SIR, 



HE folution of the problem relating to the figure of the New problems 



in fpheroidal 



lero- 



tnangles. 



T 



ear:i;, in my firft letter*, led to fome new properties of fpl 

 icial triangles, from whieli I fliall endeavour, in this, to deduce 

 a few rules, that may be applied with fuccefs in trigonometri- 

 cal furveys. It will be found that, though the general formula 

 be complex, yet, in the cafes that occur in pradice, they ad- 

 mit of being fufficiently Amplified ; for not only may the terms 

 involving the fecond and higher powers of the compreflion, 

 but alfo frequently the differences between the longitudes and 

 latitudes of the ftations, be rejected. 



1. * Having given the latitudes of two places on the furface Prob. I. Given 

 of the earth, and the length of the ftraight line or chord joining chord'of^ar"'"** 

 them, it is required to find their difference of longitude ?' Requir»d difF. 

 Let A, <P be the latitudes of the two places, D their difiance ^""S'"^* 



c 

 in fathoms, a the radiu-s of the equator in fathoms, — = I the 



a 



compreffion at the poles, d fuch that fjn \d^=.- — , and w' the 



difference of longitudes of two places, whofe latitudes are 

 X, (p, and difiance d on the fphere. Then by the formula, 

 page 16,* (he true difference of longitude w is equal to «'-{- 



Q X — =w'-{-Q^> if we rejed the higher Powersoft, in which f 

 a 



2 (fin x-fin (P)^-2 (in i d^ (fin X^-}-{in (P^-) 



col. X col. <p Im 0)' 



Now becaufe in trigonometrical lurveys, d, w', and {>-—<P) 



2 (fin x-fin (p)=^ _ 

 col. X cot. ^■iin«' 



fmall, we may take 



* Journal for May. 



f There are fome typographical and other errors in the paper 

 referred to, which I have taken notice of at the end of this letter. 



8 cof. 



