ALGEBRAIC FORMULA. 



a$3 



Thus we have two elliptic arches, reckoned from the extre- 

 mities of the axes, the difference of which is equal to a certain 

 affignable ftraight line. 



13. Having found this Very curious relation between the 

 elliptic arches AN and FD, let us next inveftigate the relation be- 

 tween the correfponding circular arches AM, HK, by means of 



the equation <p -f- (<P r ) zr -. 



Let ^ zz AM the arch of the circumfcribing circle, between 



the extremity of the tranfverfe axis, and the ordinate LNM, then 



cofij/ = fin(f), and fin 2^ = fin 2(f) ; andfihce fin 2?' =r fin 2 (*>'), 



,- r . &» i<p , r .... fin K<P) 



alio im 2P zz , . /N .— xr , — , and iin 2(P ) zz , . ^ ,— , r x ,\- , 



it follows, that to determine the relation between 9 and -f, we 

 have 



fin ip fin 2^ fin 20 fin 24, 



. .1 "■ ■ i n - . . .. » » r)Y ■— ' ■ 



(l+^0/l--^fin 1 ^ 



Let AB be the diameter of a 

 circle, of which AC the radius 

 'tzz 1 ; let the arch AD ztz 2<p, and 

 BF (taken in the oppofite femi- 

 circle) = $ty, take CP z± e\ join 

 PD, PF, CD, CF, and draw 

 DE and FG perpendicular to the 

 diameter; then DE = fin 2P, and 

 EC = cof 2 p ; alio FG h fin 2^, 

 and GC s= cof i<\>> 



'•'«+«'*— 2<r'cof2^« 





From the elements of geometry we have 



PD* j= DC * + ct * 2 + st>C . CE, 



alfoPpj =!rCi + Ci>Z "" 2i>C - CG) 



t= * -f *'* — 2(?'cof2^. 



therefore, 



