ALGEBRAIC FORMULA, 



9»% 



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

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

 affignable flraight line. 



13. Having found this very curious relation between the 

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

 tween the cori'efponding circular arches AM, HK, by means of 



the equation <p' -f {¥) =. ^. 



Let T^ = AM the arch of the circumfcribing circle, between 

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

 cof->|/ = fin(f), and fin2rj/zz{in 2(#) ; andfihce finaf'nfin 2(*'), 



ir /- > fin 2^ 1 /> r .^ fin 2(p) 



alfo fin 2P =: . , ,. , .^ , , and fm 2(P ) = , ^ , . * ,, ,, , 



(r+f') y/i— «'fin'p ' ^ ^ (i+fO/l— f'fin'(^) ' 



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

 have 



f!tl.^p fin 24- fin 21? fin 24. 



or. 



V/i+e"+2e'cof2fi^' 



(i+«')/i— ''fin'?*' (t+e'Vi— «'cof '4; ' 



Let AB be the diameter of a 

 circle, of which AC the radius 

 r= I ; let the arch AD = 2?, and 

 BF (taken in the oppofite femi- 

 circle) = i^^, take CP 1= e', join 

 PD, PF, CD, CF, and draw 

 DE and FG perpendicular to the 

 diameter; then DE = fin 2^, and 

 EC = tof 2«> ; alfo FG = fin 21^, 

 and GC = cof 2a^. 



From the elements of geometry we have " 



PD, j=DO + CP'-f 2PC.CE, 



"^/l+t" — 2«'cof2^» 



I -f- e>' 4-2^cof2^; 

 alf6j>pj=^^' + ^*"-2PC.CG, 



t= i + ^'* --2e'cof2^|/. 



therefore, 



