95 



orbit is of fmall excentrlcity. It cannot, neverthelefs, be denied that 

 Ward has obtained more merit than his flaare in it deferved. 



Let us proceed to enquire its exaft value ; for this purpofe the fol- 

 lowing theorem is neceffary, which will alfo afterwards be referred to, and 

 may be of ufe in other enquiries. 



Theorem. 



In a plane triangle, let half the bafe be e, and half the fum of the 

 fides be unity, and one of the exterior angles at the bafe = m (rad. 

 unity,) then the oppofite interior angle will = 



m — 2w, 7»+2£ J, 2OT — 2£ J, ^m+2e^ s, 4m — &c. when e islefsthan i. 

 1234 



Demonftration. Let rt=the interior oppofite angle, 

 then by Plane Trig. i-\-e : i — e : : /, ^m : i, 3a 



therefore log. /, 5^=log. t, ^m — log. i+^'+log. i — e, or making 



• • • 2 • 



e and a vary ^a — = ag o r ax i — e =— 2fj, a. 



i, faXa*, |a i— £» 



In order to find, from this equation, the value of a correfponding 

 to any values of e and ?n, in a feries afcending by the powers of e, 

 let the fuccefiive fluxions of this equation be takeir per Saltum,* mak- 

 ing e = 0, and a = m. 



then a=2eacs,m or a=2e s^am, when 'e:=e 

 2 2 



•3 •« • • " .'. 3 



a — 2f fl= — 2es, a or a = — ze s, 3m 



■^ T 



See Tranf. R. Iri/h Academy, toL 7, Mem. p. 321. Prob. 2. & 3. 



